1 00:00:00,000 --> 00:00:02,350 The following content is provided under a Creative 2 00:00:02,350 --> 00:00:03,640 Commons license. 3 00:00:03,640 --> 00:00:06,600 Your support will help MIT OpenCourseWare continue to 4 00:00:06,600 --> 00:00:09,515 offer high quality educational resources for free. 5 00:00:09,515 --> 00:00:12,815 To make a donation or to view additional materials from 6 00:00:12,815 --> 00:00:16,780 hundreds of MIT courses, visit MIT OpenCourseWare at 7 00:00:16,780 --> 00:00:18,030 ocw.mit.edu. 8 00:00:22,290 --> 00:00:27,140 PROFESSOR: I really stated everything about discrete 9 00:00:27,140 --> 00:00:29,560 coding as clearly as I could in the notes. 10 00:00:29,560 --> 00:00:33,410 I stated it again as clearly as I could in class. 11 00:00:33,410 --> 00:00:36,950 I stated it again as clearly as I could in making up 12 00:00:36,950 --> 00:00:40,470 problems that would illustrate the ideas. 13 00:00:40,470 --> 00:00:42,820 If I talked about it again it would just be totally 14 00:00:42,820 --> 00:00:43,490 repetitive. 15 00:00:43,490 --> 00:00:49,290 So, at this point, if you want to understand things better, 16 00:00:49,290 --> 00:00:52,230 you gotta come up with specific questions and I will 17 00:00:52,230 --> 00:00:56,370 be delighted to deal with them. 18 00:00:56,370 --> 00:00:58,020 So we want to go on--. 19 00:00:58,020 --> 00:01:01,580 Oh, there's one other thing I wanted to talk about. 20 00:01:01,580 --> 00:01:05,570 We're not having a new problem set out today. 21 00:01:05,570 --> 00:01:07,170 I don't think most of you would 22 00:01:07,170 --> 00:01:09,650 concentrate on it very well. 23 00:01:14,430 --> 00:01:18,590 I'll tell you what the problems are, which will be 24 00:01:18,590 --> 00:01:21,180 due on October 14th. 25 00:01:24,380 --> 00:01:25,870 I'll pass it out later. 26 00:01:28,950 --> 00:01:47,120 It's problems 1 through 7, and the end of lectures 8 to 10, 27 00:01:47,120 --> 00:01:51,060 and one other one all the way at the end, problem 26. 28 00:01:55,670 --> 00:01:59,470 So, 1, 2, 3, 4, 5, 6, 7 and 26. 29 00:02:05,270 --> 00:02:10,020 So you can get started on them whenever you choose and I'll 30 00:02:10,020 --> 00:02:15,590 pass out a traditional problem set form next time. 31 00:02:21,680 --> 00:02:25,920 Last time we started to talk about the difference between 32 00:02:25,920 --> 00:02:29,930 Reimann and Lebesgue integration. 33 00:02:29,930 --> 00:02:33,810 Most people tell me this is further into mathematics then 34 00:02:33,810 --> 00:02:35,990 I should go. 35 00:02:35,990 --> 00:02:41,430 If you agree with them after we spend a week or two on 36 00:02:41,430 --> 00:02:44,980 this, please let me know and I won't torture future 37 00:02:44,980 --> 00:02:46,940 students with it. 38 00:02:46,940 --> 00:02:51,920 My sense is that in the things we're going to be dealing with 39 00:02:51,920 --> 00:02:55,900 for most of the rest of the term, knowing a little bit 40 00:02:55,900 --> 00:03:00,340 extra about mathematics is going to save you an awful lot 41 00:03:00,340 --> 00:03:04,100 of time worrying about trivial little things that you 42 00:03:04,100 --> 00:03:06,240 shouldn't be worrying about. 43 00:03:06,240 --> 00:03:09,190 In other words, the great mathematicians of the 19th 44 00:03:09,190 --> 00:03:13,780 century who developed -- 45 00:03:13,780 --> 00:03:16,470 yeah, the 19th century, but partly the 20th century. 46 00:03:16,470 --> 00:03:20,990 These mathematicians were really engineers at heart. 47 00:03:20,990 --> 00:03:25,190 The 20th century and the 21st century, mathematics has very 48 00:03:25,190 --> 00:03:32,680 much become very separated from physics and applications. 49 00:03:32,680 --> 00:03:36,970 But in the 19th century and in the early 20th century, 50 00:03:36,970 --> 00:03:42,630 mathematicians and physicists were almost the same animals. 51 00:03:42,630 --> 00:03:45,550 You could scratch one and you'd find another one there. 52 00:03:45,550 --> 00:03:50,990 They were very much interested in dealing with real things. 53 00:03:50,990 --> 00:03:55,300 They were like the very best of mathematicians everywhere 54 00:03:55,300 --> 00:03:58,060 and the very best of engineers everywhere. 55 00:03:58,060 --> 00:04:01,830 They really wanted to make life simpler instead of making 56 00:04:01,830 --> 00:04:03,960 life more complicated. 57 00:04:03,960 --> 00:04:06,590 One way that many people express it is that 58 00:04:06,590 --> 00:04:11,100 mathematicians are lazy, and because they're lazy, they 59 00:04:11,100 --> 00:04:14,270 don't want to go through a lot of work, and therefore, they 60 00:04:14,270 --> 00:04:16,770 feel driven to simplify things. 61 00:04:16,770 --> 00:04:19,520 There's an awful lot of truth in that. 62 00:04:19,520 --> 00:04:22,450 What we're going to learn here I think will, in fact, 63 00:04:22,450 --> 00:04:27,580 simplify what you know about Fourier series and Fourier 64 00:04:27,580 --> 00:04:29,650 integrals a great deal. 65 00:04:29,650 --> 00:04:33,480 Engineers typically don't worry about those things. 66 00:04:33,480 --> 00:04:36,490 An awful lot of engineers, and unfortunately, even those who 67 00:04:36,490 --> 00:04:41,280 write books often state theorems and leave out the 68 00:04:41,280 --> 00:04:45,130 last clause of the theorem, and the last clause of most of 69 00:04:45,130 --> 00:04:49,110 those theorems the only way to make them true is to add the 70 00:04:49,110 --> 00:04:53,960 clause or not, as the case may be, at the end of it, which 71 00:04:53,960 --> 00:04:56,330 makes what they state absolutely meaningless, 72 00:04:56,330 --> 00:05:00,590 because anything you can add or not, as the case may be, 73 00:05:00,590 --> 00:05:02,640 and it becomes true at that point. 74 00:05:02,640 --> 00:05:05,550 The whole question becomes well, what are those cases 75 00:05:05,550 --> 00:05:07,450 under which it's true. 76 00:05:07,450 --> 00:05:11,930 That's what we're going to deal with a little bit here. 77 00:05:11,930 --> 00:05:16,410 We're going to say just enough so you start to understand 78 00:05:16,410 --> 00:05:20,670 these major things that cause problems, and I hope you will 79 00:05:20,670 --> 00:05:23,450 get to the point where you don't have to worry about them 80 00:05:23,450 --> 00:05:25,590 after that. 81 00:05:25,590 --> 00:05:27,890 Well, the first thing, we started talking about it last 82 00:05:27,890 --> 00:05:31,920 time, we were talking about the difference between Reimann 83 00:05:31,920 --> 00:05:34,250 integration and Lebesgue integration. 84 00:05:34,250 --> 00:05:37,430 Reimann was a great mathematician, but he came 85 00:05:37,430 --> 00:05:41,480 before all of these mathematicians who were 86 00:05:41,480 --> 00:05:45,320 following Lebesgue, and who started to deal with all of 87 00:05:45,320 --> 00:05:48,870 the problems with this classical integration, which 88 00:05:48,870 --> 00:05:54,530 started to fall apart throughout 89 00:05:54,530 --> 00:05:56,550 most of the 19th century. 90 00:05:56,550 --> 00:06:02,130 What Reimann said, well, split up the axis into equal 91 00:06:02,130 --> 00:06:06,430 intervals, then approximate the function within each 92 00:06:06,430 --> 00:06:13,410 interval, add up all of those approximate values, and then 93 00:06:13,410 --> 00:06:18,680 let this quantization over the time axis become finer and 94 00:06:18,680 --> 00:06:21,060 finer and finer. 95 00:06:21,060 --> 00:06:24,630 If you're lucky, you will come to a limit. 96 00:06:24,630 --> 00:06:27,060 You can sort of see when you get to a limit and when you 97 00:06:27,060 --> 00:06:28,310 don't get to a limit. 98 00:06:28,310 --> 00:06:31,190 If the function is smooth enough and you break it up 99 00:06:31,190 --> 00:06:34,610 finely enough, you're going to get a very good approximation, 100 00:06:34,610 --> 00:06:36,970 and if you break it up more and more finely, the 101 00:06:36,970 --> 00:06:39,470 approximation gets better and better. 102 00:06:39,470 --> 00:06:43,835 If the function is very wild, if it jumps around wildly, and 103 00:06:43,835 --> 00:06:48,190 we'll look at examples of that later, then this doesn't work. 104 00:06:48,190 --> 00:06:50,690 We'll see why, in fact, this approach does work. 105 00:06:50,690 --> 00:06:55,700 Lebesgue said, no, instead of quantizing along this axis, 106 00:06:55,700 --> 00:06:59,510 quantize on this axis. 107 00:06:59,510 --> 00:07:05,190 So he said, OK, start with a zero, quantize into epsilon, 2 108 00:07:05,190 --> 00:07:10,430 epsilon, 3 epsilon and so forth, and everybody when they 109 00:07:10,430 --> 00:07:12,150 talk about epsilon they're thinking 110 00:07:12,150 --> 00:07:14,080 about something small. 111 00:07:14,080 --> 00:07:16,720 They're also thinking about making it smaller and smaller 112 00:07:16,720 --> 00:07:20,270 and smaller and hoping that something nice happens. 113 00:07:20,270 --> 00:07:24,780 So then what he said is after you quantize this axis, start 114 00:07:24,780 --> 00:07:29,090 to look at how much of the function lies in each one of 115 00:07:29,090 --> 00:07:31,410 those little windows. 116 00:07:31,410 --> 00:07:36,290 I've drawn that out here, mu 2 is the amount of the function 117 00:07:36,290 --> 00:07:41,420 that lies between 2 epsilon and 3 epsilon. 118 00:07:41,420 --> 00:07:45,600 Now the function lies between 2 epsilon and 3 epsilon 119 00:07:45,600 --> 00:07:49,520 starting at this point, which I've labeled t1, going up to 120 00:07:49,520 --> 00:07:53,410 this point, which I've labeled t2, on over here. 121 00:07:53,410 --> 00:07:57,260 It's not in this interval until we get back to t3. 122 00:07:57,260 --> 00:08:00,460 It stays in this interval until t4. 123 00:08:00,460 --> 00:08:04,940 Lebesgue said, OK, let's say that the function is between 2 124 00:08:04,940 --> 00:08:11,790 epsilon and 3 epsilon over a region of size, t2 minus t1, 125 00:08:11,790 --> 00:08:16,310 which is this size interval, plus t4 minus t3, which is 126 00:08:16,310 --> 00:08:18,690 this size interval. 127 00:08:18,690 --> 00:08:22,800 Instead of saying size, he said size gets too confusing 128 00:08:22,800 --> 00:08:26,010 so I'll call it measure instead. 129 00:08:26,010 --> 00:08:28,450 That was the beginning of measure theory, essentially. 130 00:08:28,450 --> 00:08:30,020 Well, in fact other people talked about 131 00:08:30,020 --> 00:08:31,490 measure before Lebesgue. 132 00:08:31,490 --> 00:08:34,220 There are a lot of famous mathematicians who were all 133 00:08:34,220 --> 00:08:37,690 involved in doing this. 134 00:08:37,690 --> 00:08:41,505 Anyway, that's the basic idea of what measure 135 00:08:41,505 --> 00:08:42,690 is concerned with. 136 00:08:42,690 --> 00:08:47,370 Now, for this curve here, it's a nice smooth curve, and you 137 00:08:47,370 --> 00:08:50,160 can almost see intuitively that you're going to get the 138 00:08:50,160 --> 00:08:55,490 same thing looking at it this way or looking at it this way. 139 00:08:55,490 --> 00:08:58,810 In fact, you do. 140 00:08:58,810 --> 00:09:02,730 Anyway, what he finally wound up with is after saying what 141 00:09:02,730 --> 00:09:06,120 the measure was on each one of those little slices, how much 142 00:09:06,120 --> 00:09:08,810 of the function lay in each one of those intervals, he 143 00:09:08,810 --> 00:09:10,630 would just add them all up. 144 00:09:10,630 --> 00:09:13,160 He would add up how much of the function was in each 145 00:09:13,160 --> 00:09:16,370 slice, he would multiply how much of the function was in a 146 00:09:16,370 --> 00:09:20,480 slice by how high the slice was up, and 147 00:09:20,480 --> 00:09:21,980 then he'd get an answer. 148 00:09:21,980 --> 00:09:26,490 One difference between what he did and what Reimann did was 149 00:09:26,490 --> 00:09:29,020 that he always got a lower bound in doing it this way. 150 00:09:29,020 --> 00:09:32,310 If he was dealing with a non-negative function, his 151 00:09:32,310 --> 00:09:35,360 approximation was always a little less than what the 152 00:09:35,360 --> 00:09:39,550 function was, because anything that lay between 2 epsilon and 153 00:09:39,550 --> 00:09:43,410 3 epsilon, he would approximate it as 2 epsilon, 154 00:09:43,410 --> 00:09:46,180 which is a little less than numbers between 2 155 00:09:46,180 --> 00:09:47,760 epsilon and 3 epsilon. 156 00:09:47,760 --> 00:09:52,550 So this is a lower bound, whereas this is whatever it 157 00:09:52,550 --> 00:09:55,830 happens to be -- however you decide to approximate the 158 00:09:55,830 --> 00:10:02,960 function there, and there are lots of ways of doing it. 159 00:10:02,960 --> 00:10:05,460 Well, I won't prove any of these things, I just want to 160 00:10:05,460 --> 00:10:11,010 point them out so that when you get frustrated with this, 161 00:10:11,010 --> 00:10:13,730 you can always rely on this. 162 00:10:13,730 --> 00:10:16,220 Which says that whenever the Reimann integral exists, in 163 00:10:16,220 --> 00:10:19,290 other words, whenever the integral that you're used to 164 00:10:19,290 --> 00:10:25,260 exists, mainly, whenever it has meaning, Lebesgue integral 165 00:10:25,260 --> 00:10:27,850 gives you the same value. 166 00:10:27,850 --> 00:10:31,220 In other words, you haven't lost anything by going from 167 00:10:31,220 --> 00:10:33,910 Reimann integration to Lebesgue integration. 168 00:10:33,910 --> 00:10:36,380 You can only gain, you can't lose. 169 00:10:36,380 --> 00:10:39,340 The familiar rules for calculating Reimann integrals 170 00:10:39,340 --> 00:10:42,330 also apply for Lebesgue integrals. 171 00:10:42,330 --> 00:10:44,790 You remember what all those rules are, you probably know 172 00:10:44,790 --> 00:10:48,520 all of them better even than this fundamental definition of 173 00:10:48,520 --> 00:10:52,810 an integral, which is split up the function into tiny little 174 00:10:52,810 --> 00:10:56,940 increments, because throughout all of the courses that you've 175 00:10:56,940 --> 00:10:59,830 taken, learning about integration, learning about 176 00:10:59,830 --> 00:11:02,650 differentiation, learning about all of these things, 177 00:11:02,650 --> 00:11:05,740 what you've done for the most part is to go through 178 00:11:05,740 --> 00:11:08,720 exercises using these various rules. 179 00:11:08,720 --> 00:11:12,870 So, you know how to integrate lots of traditional functions. 180 00:11:12,870 --> 00:11:15,710 You memorized what the integral of many of them is. 181 00:11:15,710 --> 00:11:18,830 You had many ways of combining them to find out what the 182 00:11:18,830 --> 00:11:21,550 integral of many other functions is. 183 00:11:21,550 --> 00:11:25,190 If you program a computer to calculate these integrals, a 184 00:11:25,190 --> 00:11:27,600 computer can do it both ways. 185 00:11:27,600 --> 00:11:29,950 It can either use all the rules to find out what the 186 00:11:29,950 --> 00:11:34,000 value of an integral is, or it can chop things up finely and 187 00:11:34,000 --> 00:11:36,800 find out that way. 188 00:11:36,800 --> 00:11:43,120 As I said before, if you think that being able to calculate 189 00:11:43,120 --> 00:11:46,870 integrals is what engineering is about, think again. 190 00:11:46,870 --> 00:11:48,880 I told you before you could be replaced 191 00:11:48,880 --> 00:11:51,270 by a digital computer. 192 00:11:51,270 --> 00:11:52,780 It's worse than that. 193 00:11:52,780 --> 00:11:57,200 You could be replaced by your handheld Palm, 194 00:11:57,200 --> 00:11:58,790 whatever it's called. 195 00:11:58,790 --> 00:12:02,720 You can be replaced by anything. 196 00:12:02,720 --> 00:12:05,240 After a while we're going to wear little things embedded in 197 00:12:05,240 --> 00:12:06,945 our body that will tell us when we're sick 198 00:12:06,945 --> 00:12:08,330 and things like this. 199 00:12:08,330 --> 00:12:12,620 You can be replaced by those things even. 200 00:12:12,620 --> 00:12:17,230 So you really want to learn more than just that. 201 00:12:17,230 --> 00:12:20,090 For some very weird functions, the Lebesgue integral exists, 202 00:12:20,090 --> 00:12:22,470 but the Reimann integral doesn't exist. 203 00:12:22,470 --> 00:12:27,070 Why do we want to worry about weird functions? 204 00:12:27,070 --> 00:12:28,820 I want to tell you two reasons for it -- 205 00:12:28,820 --> 00:12:30,870 I told you about it last time. 206 00:12:30,870 --> 00:12:36,730 One thing is an awful lot of communication theory is 207 00:12:36,730 --> 00:12:41,050 concerned with going back and forth between the time domain 208 00:12:41,050 --> 00:12:43,820 and the frequency domain. 209 00:12:43,820 --> 00:12:47,940 When you talk about things which are straightforward in 210 00:12:47,940 --> 00:12:52,720 the time domain, often they become very, very weird in the 211 00:12:52,720 --> 00:12:54,940 frequency domain and vice versa. 212 00:12:54,940 --> 00:13:00,270 I'm going to give you a beautiful example of that 213 00:13:00,270 --> 00:13:01,330 probably in the next class. 214 00:13:01,330 --> 00:13:03,540 You can look at it -- 215 00:13:03,540 --> 00:13:06,660 it is in the appendix. 216 00:13:06,660 --> 00:13:09,630 You'll see this absolutely weird function which has a 217 00:13:09,630 --> 00:13:12,210 perfectly well-defined Fourier transform. 218 00:13:12,210 --> 00:13:15,310 Therefore, the inverse Fourier transform of this nice looking 219 00:13:15,310 --> 00:13:18,590 thing is absolutely weird. 220 00:13:18,590 --> 00:13:20,840 The other reason is even more important. 221 00:13:20,840 --> 00:13:23,490 We have to deal with random processes. 222 00:13:23,490 --> 00:13:26,430 We have to deal with noise functions, which are 223 00:13:26,430 --> 00:13:29,460 continuously varying functions of time. 224 00:13:29,460 --> 00:13:32,660 We have to deal with what we transmit, which is 225 00:13:32,660 --> 00:13:36,270 continuously varying functions of time. 226 00:13:36,270 --> 00:13:39,530 Just like when we were dealing with sources, we said if we 227 00:13:39,530 --> 00:13:44,250 want to compress a source it's not enough to think about what 228 00:13:44,250 --> 00:13:46,110 one source sequence is. 229 00:13:46,110 --> 00:13:49,060 We have to think about it probabalistically. 230 00:13:49,060 --> 00:13:55,260 Namely, we have to model these functions as sample values of 231 00:13:55,260 --> 00:13:59,500 what we call a stochastic process or a random process. 232 00:13:59,500 --> 00:14:02,150 In other words, when we start talking about that, these 233 00:14:02,150 --> 00:14:06,040 functions which already look complicated just become sample 234 00:14:06,040 --> 00:14:10,000 points in this much bigger space where we're dealing with 235 00:14:10,000 --> 00:14:11,640 random processes. 236 00:14:11,640 --> 00:14:15,090 Now, when you're dealing with sample points of these random 237 00:14:15,090 --> 00:14:20,460 processes, weirdness just crops up as a necessary part 238 00:14:20,460 --> 00:14:21,830 of all of this. 239 00:14:21,830 --> 00:14:26,750 You can't define a random process which consists of only 240 00:14:26,750 --> 00:14:28,540 non-weird things. 241 00:14:28,540 --> 00:14:31,430 If you do you get something that doesn't work very well, 242 00:14:31,430 --> 00:14:33,350 it doesn't do much for you. 243 00:14:33,350 --> 00:14:37,200 People do that all the time, but you run out of steam with 244 00:14:37,200 --> 00:14:39,250 it very, very quickly. 245 00:14:39,250 --> 00:14:42,440 So for both reasons we have to be able to 246 00:14:42,440 --> 00:14:44,090 deal with weird things. 247 00:14:44,090 --> 00:14:47,940 The most ideal thing is to be able to deal with weird things 248 00:14:47,940 --> 00:14:51,750 and get rid of them without even thinking about them. 249 00:14:51,750 --> 00:14:54,890 What's the nice thing about Lebesgue theory? 250 00:14:54,890 --> 00:14:57,270 It let's you get rid of all the weirdness without even 251 00:14:57,270 --> 00:14:59,690 thinking about it. 252 00:14:59,690 --> 00:15:02,550 But in order to understand it to start with so we get rid of 253 00:15:02,550 --> 00:15:06,440 all that weird stuff, we have to understand just a little 254 00:15:06,440 --> 00:15:09,710 bit about what it's about to start with. 255 00:15:09,710 --> 00:15:16,610 So that's where we're going. 256 00:15:16,610 --> 00:15:21,790 Well, in this picture -- let me put the picture 257 00:15:21,790 --> 00:15:26,420 back up again -- 258 00:15:26,420 --> 00:15:28,250 I sort of sluffed over something. 259 00:15:28,250 --> 00:15:31,850 For a nice, well-behaved function, it's easy to find 260 00:15:31,850 --> 00:15:37,020 out what the measure of a function is. 261 00:15:37,020 --> 00:15:41,560 Namely, what the measure is of the set of values where you're 262 00:15:41,560 --> 00:15:45,270 in some tiny little interval, and that's straightforward, it 263 00:15:45,270 --> 00:15:49,310 was just the sum of a bunch of intervals. 264 00:15:52,290 --> 00:15:56,000 But now we come to the clincher, which is we want to 265 00:15:56,000 --> 00:15:59,380 be able to deal with weird functions too, and for weird 266 00:15:59,380 --> 00:16:05,540 functions this measure, the set of values of time in which 267 00:16:05,540 --> 00:16:08,780 the function is in some tiny little slice here, the measure 268 00:16:08,780 --> 00:16:11,960 of that is going to become rather complicated. 269 00:16:11,960 --> 00:16:14,760 Therefore, we have to understand how to find the 270 00:16:14,760 --> 00:16:17,550 measure of some pretty weird sets. 271 00:16:17,550 --> 00:16:22,730 So, Lebesgue went on and said OK, I'll define how to find 272 00:16:22,730 --> 00:16:25,160 measure of things. 273 00:16:25,160 --> 00:16:26,910 It's all very intuitive. 274 00:16:26,910 --> 00:16:31,530 If you have any real numbers, a and b -- 275 00:16:31,530 --> 00:16:33,990 if I have two real numbers I might as well say one of them 276 00:16:33,990 --> 00:16:37,120 was less than or equal to the other one, so a is less than 277 00:16:37,120 --> 00:16:38,620 or equal to b. 278 00:16:38,620 --> 00:16:40,990 I want to include minus infinity and 279 00:16:40,990 --> 00:16:42,510 plus infinity here. 280 00:16:42,510 --> 00:16:46,620 When we say the set of real numbers, we don't include 281 00:16:46,620 --> 00:16:49,650 minus infinity and plus infinity. 282 00:16:49,650 --> 00:16:52,800 When we talk about the extended set of real numbers, 283 00:16:52,800 --> 00:16:56,370 we doing include minus infinity and plus infinity, 284 00:16:56,370 --> 00:16:58,780 Here for the most part when we're dealing with measure 285 00:16:58,780 --> 00:17:01,990 theory, we really want to include minus infinity and 286 00:17:01,990 --> 00:17:06,190 plus infinity also because they make things easier to 287 00:17:06,190 --> 00:17:07,430 talk about. 288 00:17:07,430 --> 00:17:13,140 So what Lebesgue said is for any interval the set of points 289 00:17:13,140 --> 00:17:17,910 which lie between a and b, and he started out with the open 290 00:17:17,910 --> 00:17:21,290 interval -- namely the set of points not including a and not 291 00:17:21,290 --> 00:17:25,690 including b, but including all the real numbers in between -- 292 00:17:25,690 --> 00:17:29,510 the measure of that is what we said before, and none of you 293 00:17:29,510 --> 00:17:30,950 objected to it. 294 00:17:30,950 --> 00:17:33,950 The measure of an interval ought to be the size of the 295 00:17:33,950 --> 00:17:37,900 interval, because after all, all Lebesgue was doing was 296 00:17:37,900 --> 00:17:41,430 taking size and putting another name on it because we 297 00:17:41,430 --> 00:17:44,930 all thought of it as being size. 298 00:17:44,930 --> 00:17:49,890 So the measure of the interval ab is b minus a. 299 00:17:49,890 --> 00:17:52,390 Now as engineers we all know that it doesn't really make 300 00:17:52,390 --> 00:17:55,050 any difference whether you include a or you don't include 301 00:17:55,050 --> 00:17:58,170 a when we're trying to find the size of an interval. 302 00:17:58,170 --> 00:18:03,880 So, he said OK, the size of that interval is b minus a, 303 00:18:03,880 --> 00:18:08,370 whether or not you include either of the end points. 304 00:18:08,370 --> 00:18:12,060 Then he went on to say something else, the measure of 305 00:18:12,060 --> 00:18:15,740 a countable union of disjoint intervals is the sum of the 306 00:18:15,740 --> 00:18:17,540 measure of each interval. 307 00:18:22,340 --> 00:18:26,040 We already said when we were trying to figure out what the 308 00:18:26,040 --> 00:18:31,020 Lebesgue integral was, that if you had several intervals, the 309 00:18:31,020 --> 00:18:34,845 measure of the entire interval, namely the measure 310 00:18:34,845 --> 00:18:39,330 of the union of those intervals ought to be the sum 311 00:18:39,330 --> 00:18:41,990 of the measure of each interval. 312 00:18:41,990 --> 00:18:45,890 So all he's adding here is let's go on and go to the 313 00:18:45,890 --> 00:18:51,410 limit and talk about a countably infinite number of 314 00:18:51,410 --> 00:18:55,100 intervals and use the same definition. 315 00:18:55,100 --> 00:18:58,840 Now there's one nice thing about that and what is it? 316 00:18:58,840 --> 00:19:06,560 If we take the sum of a countable set of things, each 317 00:19:06,560 --> 00:19:09,110 of those things is non-negative. 318 00:19:09,110 --> 00:19:13,300 So what happens when we take the sum of a bunch of 319 00:19:13,300 --> 00:19:14,780 non-negative numbers? 320 00:19:19,170 --> 00:19:21,120 There are only two possibilities when you take 321 00:19:21,120 --> 00:19:26,430 the sum of even a countable set of non-negative numbers 322 00:19:26,430 --> 00:19:27,210 and what are they? 323 00:19:27,210 --> 00:19:31,980 AUDIENCE: Either to the limit or it goes to infinity? 324 00:19:31,980 --> 00:19:36,040 PROFESSOR: It goes to a limit or it goes to infinity, yes. 325 00:19:36,040 --> 00:19:39,090 Lebesgue said well, let's say if it goes to infinity that's 326 00:19:39,090 --> 00:19:41,230 a limit also. 327 00:19:41,230 --> 00:19:43,620 So, in other words, there are only two possibilities -- the 328 00:19:43,620 --> 00:19:46,590 sum is finite, you can find the sum, 329 00:19:46,590 --> 00:19:47,990 and the sum if infinite. 330 00:19:47,990 --> 00:19:50,170 Nothing else can happen. 331 00:19:50,170 --> 00:19:52,760 So that's nice. 332 00:19:52,760 --> 00:19:55,090 Then Lebesgue said one other thing. 333 00:19:55,090 --> 00:19:58,800 A bunch of mathematicians trying to deal with this did 334 00:19:58,800 --> 00:20:03,020 different things when they were trying to deal with very 335 00:20:03,020 --> 00:20:05,490 small sets. 336 00:20:05,490 --> 00:20:08,230 Some of them said well these very small sets aren't 337 00:20:08,230 --> 00:20:12,980 measurable, others, and part of Lebesgue's genius was that 338 00:20:12,980 --> 00:20:17,200 he said if you can take a set of points and you can put them 339 00:20:17,200 --> 00:20:20,740 inside another set of points, which doesn't amount to 340 00:20:20,740 --> 00:20:25,800 anything, then this smaller set of points should have a 341 00:20:25,800 --> 00:20:27,830 measure which is less than or equal to the 342 00:20:27,830 --> 00:20:30,980 bigger set of points. 343 00:20:30,980 --> 00:20:37,240 So that that says that any subset of something that has 344 00:20:37,240 --> 00:20:40,860 zero measure also ought to have zero measure. 345 00:20:40,860 --> 00:20:42,750 Now when we look at some examples you'll see that 346 00:20:42,750 --> 00:20:46,030 that's not quite as obvious as it seems. 347 00:20:46,030 --> 00:20:47,810 But anyway, he said that. 348 00:20:47,810 --> 00:20:52,360 An even nicer way to put that is when you're talking about 349 00:20:52,360 --> 00:20:59,010 weird intervals, try to cover that weird set 350 00:20:59,010 --> 00:21:03,200 with a bunch of intervals. 351 00:21:03,200 --> 00:21:06,180 If you can cover it with a bunch of intervals in 352 00:21:06,180 --> 00:21:09,640 different ways and the measure of the bunch of intervals, the 353 00:21:09,640 --> 00:21:13,550 countable set of intervals, is you can make it arbitrarily 354 00:21:13,550 --> 00:21:18,060 small, then we also say that this set has measure zero. 355 00:21:18,060 --> 00:21:23,750 So any time you have a set s and you can cover it with 356 00:21:23,750 --> 00:21:27,780 intervals which have arbitrarily small measure, 357 00:21:27,780 --> 00:21:30,300 namely we can make the measure of those intervals as small as 358 00:21:30,300 --> 00:21:34,910 we want to make them, we say bingo, s has zero measure. 359 00:21:34,910 --> 00:21:38,000 Now that's going to be very nice, because it let's us get 360 00:21:38,000 --> 00:21:40,430 rid of an awful lot of things. 361 00:21:40,430 --> 00:21:45,690 Because any time we have some set which has zero measure in 362 00:21:45,690 --> 00:21:48,760 this sense, when we look back at what we did with the 363 00:21:48,760 --> 00:21:53,140 Lebesgue integral, it says good, forget about it. 364 00:21:53,140 --> 00:21:55,700 If it has zero measure it doesn't come into 365 00:21:55,700 --> 00:21:58,620 the integral at all. 366 00:21:58,620 --> 00:22:00,320 That's exactly the thing we want. 367 00:22:00,320 --> 00:22:01,880 We'd like to get rid of all that stuff. 368 00:22:04,620 --> 00:22:07,820 So if we're going to get rid of it we have these sets which 369 00:22:07,820 --> 00:22:11,450 have measure zero, they don't amount to anything, and we're 370 00:22:11,450 --> 00:22:15,270 not going to worry about them. 371 00:22:15,270 --> 00:22:18,930 Let's do an example. 372 00:22:18,930 --> 00:22:21,960 It's a famous example. 373 00:22:21,960 --> 00:22:25,750 What about the set of rationals which lie between 374 00:22:25,750 --> 00:22:28,010 zero and 1? 375 00:22:28,010 --> 00:22:32,020 Well, rational numbers are numbers where there's an 376 00:22:32,020 --> 00:22:36,790 integer numerator and an integer denominator, and 377 00:22:36,790 --> 00:22:40,270 people have shown that there are numbers which are 378 00:22:40,270 --> 00:22:43,060 approximated by rational numbers but they're not 379 00:22:43,060 --> 00:22:46,000 rational numbers. 380 00:22:46,000 --> 00:22:49,210 You can take that set of rational numbers and you can 381 00:22:49,210 --> 00:22:50,330 order them. 382 00:22:50,330 --> 00:22:54,960 The way I've ordered them here, you can't order them in 383 00:22:54,960 --> 00:22:56,610 terms of how big they are. 384 00:22:56,610 --> 00:23:00,400 You can't start with the smallest rational number which 385 00:23:00,400 --> 00:23:04,830 is greater than zero because there isn't any such thing. 386 00:23:04,830 --> 00:23:10,170 Whatever rational number you find which is greater than 387 00:23:10,170 --> 00:23:15,580 zero, I can take a half of it, that's a rational number, and 388 00:23:15,580 --> 00:23:17,460 that also is greater than zero and it's 389 00:23:17,460 --> 00:23:19,600 smaller than your number. 390 00:23:19,600 --> 00:23:21,840 If you don't like me getting the better of you, you can 391 00:23:21,840 --> 00:23:24,770 then take half of that and come up with a smaller number 392 00:23:24,770 --> 00:23:27,820 than I could find, and we can go back and forth on this as 393 00:23:27,820 --> 00:23:30,530 long as we want. 394 00:23:30,530 --> 00:23:33,310 So the way we have to order these is a little 395 00:23:33,310 --> 00:23:34,830 more subtle than that. 396 00:23:34,830 --> 00:23:38,780 Here we're going to order them in terms of first the size of 397 00:23:38,780 --> 00:23:42,330 the denominator and next, the size of the numerator. 398 00:23:42,330 --> 00:23:45,690 So there's only one fraction in this interval -- 399 00:23:50,980 --> 00:23:52,290 oh, I screwed that up. 400 00:23:52,290 --> 00:23:55,960 I really wanted to look at the set of rational in the open 401 00:23:55,960 --> 00:24:00,123 interval between zero and 1, because I don't want zero in 402 00:24:00,123 --> 00:24:03,240 it, and I don't want 1 in it, but that's a triviality. 403 00:24:03,240 --> 00:24:06,350 You can put zero and 1 in or not. 404 00:24:06,350 --> 00:24:08,770 So, anyway, we'll leave zero and 1 out. 405 00:24:08,770 --> 00:24:10,830 So I start out with 1/2 -- 406 00:24:10,830 --> 00:24:14,890 that's the only rational number strictly between zero 407 00:24:14,890 --> 00:24:17,560 and 1 with a denominator of 2. 408 00:24:17,560 --> 00:24:23,520 Then look at the numbers which have a denominator of 3, so I 409 00:24:23,520 --> 00:24:25,830 have 1/3 and 2/3. 410 00:24:25,830 --> 00:24:29,570 I look then at the rational numbers which have a 411 00:24:29,570 --> 00:24:31,130 denominator of 4. 412 00:24:31,130 --> 00:24:34,620 I have 1/4, I have to leave out 2/4, because that's the 413 00:24:34,620 --> 00:24:38,860 same as 1/2 which I've already counted, so I have 3/4, Then I 414 00:24:38,860 --> 00:24:44,900 go on to 1/5, 2/5, 3/5, 4/5 and so forth. 415 00:24:44,900 --> 00:24:50,230 In doing this I have actually counted what the integers are. 416 00:24:50,230 --> 00:24:53,600 Namely, whenever you can take a set and put it into 417 00:24:53,600 --> 00:24:57,020 correspondence with the integers, you're showing that 418 00:24:57,020 --> 00:24:58,840 it's countable. 419 00:24:58,840 --> 00:25:00,760 I've even labeled what the counting is. 420 00:25:00,760 --> 00:25:05,190 A sub 1 is 1/2, A sub 2 is 1/3 and so forth. 421 00:25:05,190 --> 00:25:12,500 Now, I want to stick this set inside of a set of intervals, 422 00:25:12,500 --> 00:25:14,220 and that's pretty easy. 423 00:25:14,220 --> 00:25:20,130 I stick A sub i inside of the closed interval, which is A 424 00:25:20,130 --> 00:25:23,460 sub i on one side and A sub i on the other side. 425 00:25:23,460 --> 00:25:25,820 So I'm sticking it inside of an interval 426 00:25:25,820 --> 00:25:29,310 which has zero measure. 427 00:25:29,310 --> 00:25:32,010 Then what I'm going to do is I'm going to add up all of 428 00:25:32,010 --> 00:25:32,560 those things. 429 00:25:32,560 --> 00:25:35,220 Whenever you add up an infinite number of things, you 430 00:25:35,220 --> 00:25:38,240 really have to add up a finite number of them and then look 431 00:25:38,240 --> 00:25:41,130 at what happens when you go to the limit. 432 00:25:41,130 --> 00:25:46,980 So I add up all of these zeroes, and when I add up n of 433 00:25:46,980 --> 00:25:48,570 them I get zero. 434 00:25:48,570 --> 00:25:52,460 I continue to add zeroes, I continue to have zero, and the 435 00:25:52,460 --> 00:25:55,370 limit is zero. 436 00:25:55,370 --> 00:25:59,380 Now here's where the mathematicians have pulled a 437 00:25:59,380 --> 00:26:04,140 very clever swindle on you, because what they're saying is 438 00:26:04,140 --> 00:26:07,640 that infinity times zero is zero. 439 00:26:07,640 --> 00:26:12,510 But they're saying it in a very precise way. 440 00:26:12,510 --> 00:26:16,760 But anyway, since we said it in a very precise way, 441 00:26:16,760 --> 00:26:21,740 infinity times zero here is zero, the way we've said it. 442 00:26:21,740 --> 00:26:25,640 But somehow that's not very satisfying. 443 00:26:25,640 --> 00:26:28,990 I mean it really looks like we've cheated. 444 00:26:28,990 --> 00:26:31,320 So let's go on and do this in a different way. 445 00:26:34,410 --> 00:26:37,670 What we're going to do now is for each of these rational 446 00:26:37,670 --> 00:26:41,330 numbers, we're going to put a little hat around them, a 447 00:26:41,330 --> 00:26:43,640 little rectangular hat around them. 448 00:26:43,640 --> 00:26:50,150 Namely a little interval which includes A sub i, and which 449 00:26:50,150 --> 00:26:55,040 also goes delta over 2 that way, delta over 2 this way, 450 00:26:55,040 --> 00:26:57,650 and also multiplies that by -- 451 00:27:07,270 --> 00:27:10,500 my computer knew I was going to talk about computers 452 00:27:10,500 --> 00:27:13,360 replacing people, so it made some mistakes to 453 00:27:13,360 --> 00:27:15,160 get even with me. 454 00:27:15,160 --> 00:27:19,210 So this is delta times 2 to the minus i minus 1, and delta 455 00:27:19,210 --> 00:27:22,030 times 2 to the minus i minus 1. 456 00:27:22,030 --> 00:27:26,080 In other words, you take 1/2 and you put a pretty big 457 00:27:26,080 --> 00:27:27,330 interval around it. 458 00:27:27,330 --> 00:27:32,390 You take 1/3, you put a smaller interval around it. 459 00:27:32,390 --> 00:27:37,480 2/3 you put a smaller interval around that and so forth. 460 00:27:37,480 --> 00:27:41,670 Well, these intervals here are going to overlap. 461 00:27:41,670 --> 00:27:46,290 But anyway, the union of two overlapping open intervals is 462 00:27:46,290 --> 00:27:49,230 an open interval which has a smaller measure 463 00:27:49,230 --> 00:27:50,120 than with two of them. 464 00:27:50,120 --> 00:27:54,980 In other words, if you take this, the measure of this and 465 00:27:54,980 --> 00:27:57,990 add it to the measure of this, the union of this and 466 00:27:57,990 --> 00:28:00,030 this is just this. 467 00:28:03,630 --> 00:28:07,020 We don't have to be very sophisticated to see that this 468 00:28:07,020 --> 00:28:12,800 length is less than this length plus length, because 469 00:28:12,800 --> 00:28:15,750 I've double counted things here. 470 00:28:15,750 --> 00:28:19,070 So anyway, when I do this I add up the measure of all of 471 00:28:19,070 --> 00:28:23,140 these things and I get delta, and I then let delta get as 472 00:28:23,140 --> 00:28:25,500 small as I want to. 473 00:28:25,500 --> 00:28:29,470 Again, I find that the measure of the rationals is zero. 474 00:28:33,840 --> 00:28:38,990 Now if you don't like this you should be very happy with 475 00:28:38,990 --> 00:28:44,530 yourselves, because I've struggled with this for years 476 00:28:44,530 --> 00:28:48,780 and it's not intuitive, because with these intervals 477 00:28:48,780 --> 00:28:51,910 I'm putting in here, no matter how small I make that 478 00:28:51,910 --> 00:28:55,600 interval, there's an infinite number of rational numbers 479 00:28:55,600 --> 00:28:58,560 which are in that interval. 480 00:28:58,560 --> 00:29:00,820 In other words, the thing we're trying to do is to 481 00:29:00,820 --> 00:29:10,420 separate the interval 0,1 into some union of intervals which 482 00:29:10,420 --> 00:29:15,400 don't amount to anything but which include all of the 483 00:29:15,400 --> 00:29:17,660 rational numbers. 484 00:29:17,660 --> 00:29:21,700 Somehow this argument is a little bit bogus because no 485 00:29:21,700 --> 00:29:25,620 matter what number I look at between zero and 1, there are 486 00:29:25,620 --> 00:29:30,250 rational numbers arbitrarily close to it. 487 00:29:30,250 --> 00:29:35,030 In other words, what's going on here is strictly a matter 488 00:29:35,030 --> 00:29:39,010 of which order we take limits in, and that's what makes the 489 00:29:39,010 --> 00:29:42,160 argument subtle. 490 00:29:42,160 --> 00:29:44,920 But anyway, that is a perfectly sound 491 00:29:44,920 --> 00:29:46,130 mathematical argument. 492 00:29:46,130 --> 00:29:48,060 You can't get around it. 493 00:29:48,060 --> 00:29:50,760 It's why people objected to what Lebesgue was doing for a 494 00:29:50,760 --> 00:29:54,370 long time, because it wasn't intuitive to them either. 495 00:29:54,370 --> 00:29:59,260 It was intuitive to Lebesgue, and finally it's become 496 00:29:59,260 --> 00:30:03,080 intuitive to everyone, but not really intuitive. 497 00:30:03,080 --> 00:30:06,970 It's just that mathematicians have heard it so many times 498 00:30:06,970 --> 00:30:09,380 that they believe it. 499 00:30:09,380 --> 00:30:13,360 I mean one of the problems with any society is that if 500 00:30:13,360 --> 00:30:15,810 you tell people things often enough they 501 00:30:15,810 --> 00:30:18,760 start to believe them. 502 00:30:18,760 --> 00:30:21,720 Unfortunately, that's true in mathematics, too. 503 00:30:21,720 --> 00:30:25,250 Fortunately in mathematics we have proofs of things, so that 504 00:30:25,250 --> 00:30:27,860 when somebody is telling you something again and again 505 00:30:27,860 --> 00:30:30,780 which is false, there are always people who will look at 506 00:30:30,780 --> 00:30:33,670 it and say no, that's not true. 507 00:30:33,670 --> 00:30:37,500 Whereas in other cases, not necessarily. 508 00:30:40,230 --> 00:30:45,300 There are also uncountable sets with measure zero. 509 00:30:45,300 --> 00:30:48,260 For those of you who are already sort of overwhelmed by 510 00:30:48,260 --> 00:30:51,890 this, why don't you go to sleep for three minutes, it'll 511 00:30:51,890 --> 00:30:54,960 only take three minutes to talk about this, but 512 00:30:54,960 --> 00:30:58,450 it's kind of cute. 513 00:30:58,450 --> 00:31:01,920 In the set I want to talk about something closely 514 00:31:01,920 --> 00:31:06,990 related to what people call the Cantor set, but it's a 515 00:31:06,990 --> 00:31:08,590 little bit simpler than that. 516 00:31:08,590 --> 00:31:11,220 So what I'd like to do, and this is 517 00:31:11,220 --> 00:31:13,350 already familiar to you. 518 00:31:13,350 --> 00:31:17,850 I can take numbers between zero and 1 and I can represent 519 00:31:17,850 --> 00:31:19,460 them in a binary expansion. 520 00:31:19,460 --> 00:31:22,820 I can also represent them in a ternary expansion. 521 00:31:22,820 --> 00:31:26,140 I can also represent them in a decimal expansion, which is 522 00:31:26,140 --> 00:31:28,770 what you've been doing since you were three years old or 523 00:31:28,770 --> 00:31:32,110 five years older or whenever you started doing this. 524 00:31:32,110 --> 00:31:35,500 Well, ternary is simpler than decimal, so you could have 525 00:31:35,500 --> 00:31:38,960 done this a year before you started to deal with decimal 526 00:31:38,960 --> 00:31:41,590 expansions. 527 00:31:41,590 --> 00:31:45,170 So I want to look at all of the ternary expansions. 528 00:31:45,170 --> 00:31:49,870 Each real number corresponds to a ternary expansion, which 529 00:31:49,870 --> 00:31:55,480 is an infinite sequence of numbers each of which are 530 00:31:55,480 --> 00:31:56,910 zero, 1 or 2. 531 00:31:59,880 --> 00:32:04,170 Now, what I'm going to do is I'm going to remove all of the 532 00:32:04,170 --> 00:32:09,750 sequences which contain any 1's in them at all. 533 00:32:09,750 --> 00:32:13,510 Now it's not immediately clear what that's going to do, but 534 00:32:13,510 --> 00:32:15,150 think of it this way. 535 00:32:15,150 --> 00:32:18,790 If I first look at the sequences, I'm going to remove 536 00:32:18,790 --> 00:32:21,340 all the sequences which start with 1. 537 00:32:21,340 --> 00:32:24,330 So the sequences which start with 1 and have anything else 538 00:32:24,330 --> 00:32:29,820 after them, that's really the interval that starts at 1/3 539 00:32:29,820 --> 00:32:34,760 and ends at 2/3, because that's really what starting 540 00:32:34,760 --> 00:32:36,510 with 1 means. 541 00:32:36,510 --> 00:32:38,130 This is what we talked about when we talked about 542 00:32:38,130 --> 00:32:41,260 approximating binary numbers also, if you remember, is the 543 00:32:41,260 --> 00:32:43,790 way we proved the Kraft inequality. 544 00:32:43,790 --> 00:32:46,985 It was the same idea. 545 00:32:46,985 --> 00:32:53,800 The sequences which have a 1 in the second position, when 546 00:32:53,800 --> 00:32:59,180 we remove them we're removing the interval from 1/9 to 2/9. 547 00:32:59,180 --> 00:33:03,390 We're also removing the interval from 7/9 to 8/9. 548 00:33:03,390 --> 00:33:07,240 We're also removing the 4/9 to 5/9 interval, but we removed 549 00:33:07,240 --> 00:33:08,650 that before. 550 00:33:08,650 --> 00:33:17,130 So we wind up with something which is now -- we've taken 551 00:33:17,130 --> 00:33:20,990 out this, we've now taken out this, and 552 00:33:20,990 --> 00:33:22,200 we've taken out this. 553 00:33:22,200 --> 00:33:29,610 So the only thing left is this and this and 554 00:33:29,610 --> 00:33:33,130 this and this, right? 555 00:33:33,130 --> 00:33:35,620 And we've removed everything else. 556 00:33:35,620 --> 00:33:38,390 We keep on doing this. 557 00:33:38,390 --> 00:33:44,110 Well, each time we remove one of these, all of the numbers 558 00:33:44,110 --> 00:33:50,280 that's -- when I do this n times, the first time I go 559 00:33:50,280 --> 00:33:54,340 through this process, I removed 1/3 of the numbers, 560 00:33:54,340 --> 00:33:58,300 I'm left with 2/3 of the interval. 561 00:33:58,300 --> 00:34:02,010 When I remove everything that has a 1 in the second 562 00:34:02,010 --> 00:34:06,840 position, I'm down with 2/3 squared. 563 00:34:06,840 --> 00:34:11,540 When I remove everything which has a 1 in the third position, 564 00:34:11,540 --> 00:34:14,370 I'm down to 2/3 cubed. 565 00:34:14,370 --> 00:34:16,480 When I keep on doing this forever, what happens 566 00:34:16,480 --> 00:34:18,820 to 2/3 to the n? 567 00:34:18,820 --> 00:34:22,990 Well, 2/3 to the n goes to zero. 568 00:34:22,990 --> 00:34:28,370 In other words, I have removed an interval, I have removed a 569 00:34:28,370 --> 00:34:32,030 set, the measure 1, and therefore I'm left with a set 570 00:34:32,030 --> 00:34:33,440 of measure zero. 571 00:34:33,440 --> 00:34:35,290 You can see this happening. 572 00:34:35,290 --> 00:34:40,220 I mean you only have diddly left here and I keep cutting 573 00:34:40,220 --> 00:34:41,910 away at it. 574 00:34:41,910 --> 00:34:45,600 So less and less gets left. 575 00:34:45,600 --> 00:34:52,040 But what we now have is all sequences, all infinite 576 00:34:52,040 --> 00:34:57,700 sequences of zeroes and 2's. 577 00:34:57,700 --> 00:35:01,760 So I'm left with all binary sequences except instead of 578 00:35:01,760 --> 00:35:05,480 binary sequences with zeros and 1's, I now have binary 579 00:35:05,480 --> 00:35:08,880 sequences with zeros and 2's. 580 00:35:08,880 --> 00:35:16,150 How many binary sequences are there when I continue forever? 581 00:35:16,150 --> 00:35:19,070 Well, you know they're an uncountable number, because if 582 00:35:19,070 --> 00:35:22,810 I take all the numbers between zero and 1, I represent them 583 00:35:22,810 --> 00:35:25,460 in binary zeroes and 1's, I have an 584 00:35:25,460 --> 00:35:29,540 uncountable number of them. 585 00:35:29,540 --> 00:35:33,480 Well, because I have to have an uncountable number because 586 00:35:33,480 --> 00:35:36,710 we already showed that any countable set doesn't amount 587 00:35:36,710 --> 00:35:38,230 to anything. 588 00:35:38,230 --> 00:35:41,770 Countable sets are diddly. 589 00:35:41,770 --> 00:35:44,270 Countable sets all just go away. 590 00:35:44,270 --> 00:35:48,430 So, anything which gets left has to be uncountable. 591 00:35:48,430 --> 00:35:51,260 Again, people had to worry about this for a long time. 592 00:35:51,260 --> 00:35:55,380 But anyway, this gives you an uncountable set which has 593 00:35:55,380 --> 00:35:56,630 measure zero. 594 00:35:59,410 --> 00:36:02,360 So, back to measurable functions. 595 00:36:02,360 --> 00:36:05,830 I'm going to get off of mathematics relatively soon, 596 00:36:05,830 --> 00:36:08,730 but we need at least this much to figure out 597 00:36:08,730 --> 00:36:10,030 what's going on here. 598 00:36:10,030 --> 00:36:12,250 We say that a function is measurable. 599 00:36:12,250 --> 00:36:15,740 Before we were only talking about sets of numbers being 600 00:36:15,740 --> 00:36:18,240 measurable. 601 00:36:18,240 --> 00:36:21,950 We had to talk about sets of numbers being measurable 602 00:36:21,950 --> 00:36:26,330 because we were interested in the question of what's the set 603 00:36:26,330 --> 00:36:32,440 of times for which a function lies between 2 epsilon and 3 604 00:36:32,440 --> 00:36:35,370 epsilon, for example. 605 00:36:35,370 --> 00:36:39,530 What we said is we can say a great deal about that because 606 00:36:39,530 --> 00:36:44,120 we can not only add up a bunch of intervals, we can also add 607 00:36:44,120 --> 00:36:47,100 up a countable bunch of intervals, and we can also get 608 00:36:47,100 --> 00:36:50,120 rid of anything which is negligible. 609 00:36:50,120 --> 00:36:56,220 So, a function is measurable if the set of t, such that u 610 00:36:56,220 --> 00:36:58,970 of t lies between these two points is 611 00:36:58,970 --> 00:37:00,870 measurable for each interval. 612 00:37:00,870 --> 00:37:05,230 In other words, if no matter how I split up this interval, 613 00:37:05,230 --> 00:37:13,660 if no matter what slice I look at, the set of times over 614 00:37:13,660 --> 00:37:17,820 which the function lies in there is measurable. 615 00:37:17,820 --> 00:37:21,010 That's what a measurable function is. 616 00:37:21,010 --> 00:37:25,000 Everybody understand what I just said? 617 00:37:25,000 --> 00:37:27,520 Let me try to say it once more. 618 00:37:27,520 --> 00:37:33,300 A function is measurable if for every two values, say 3 619 00:37:33,300 --> 00:37:38,840 epsilon and 2 epsilon, if the set of values t for which the 620 00:37:38,840 --> 00:37:43,340 function lies between 2 epsilon and 3 epsilon, if that 621 00:37:43,340 --> 00:37:46,030 set is measurable. 622 00:37:46,030 --> 00:37:49,550 In other words, that's the set we were talking about before 623 00:37:49,550 --> 00:37:53,640 which went from here to here, and which went 624 00:37:53,640 --> 00:37:55,660 from here to there. 625 00:37:55,660 --> 00:37:58,700 In this case for this very simple function, that's just 626 00:37:58,700 --> 00:38:00,520 the sum of two intervals. 627 00:38:00,520 --> 00:38:04,370 If I make the function wiggle a great deal more, it's the 628 00:38:04,370 --> 00:38:07,850 sum of a lot more intervals. 629 00:38:07,850 --> 00:38:12,075 So, we say the function is measurable if all of the sets 630 00:38:12,075 --> 00:38:13,560 are measurable. 631 00:38:13,560 --> 00:38:16,900 Now, what I'm going to do is when I'm trying to define this 632 00:38:16,900 --> 00:38:19,200 integral, I'm going to have to go to 633 00:38:19,200 --> 00:38:22,450 smaller and smaller intervals. 634 00:38:22,450 --> 00:38:25,380 Let's start out with epsilon, 2 epsilon, 3 635 00:38:25,380 --> 00:38:27,240 epsilon and so forth. 636 00:38:27,240 --> 00:38:29,360 Let's look at a non-negative function--. 637 00:38:31,970 --> 00:38:32,640 Yeah? 638 00:38:32,640 --> 00:38:35,928 AUDIENCE: Maybe I missed something, but could you tell 639 00:38:35,928 --> 00:38:40,300 me [UNINTELLIGIBLE] definition for a set, if measurable. 640 00:38:40,300 --> 00:38:41,870 PROFESSOR: What's the definition for a set is 641 00:38:41,870 --> 00:38:44,710 measurable. 642 00:38:44,710 --> 00:38:49,350 I didn't really say, and that's good. 643 00:38:49,350 --> 00:38:51,650 I gave you a bunch of conditions under which a set 644 00:38:51,650 --> 00:38:55,560 is measurable, and if I have enough conditions for which 645 00:38:55,560 --> 00:38:59,710 it's measurable then I don't have to worry about--. 646 00:39:03,060 --> 00:39:07,010 I said that it is measurable under all of these conditions. 647 00:39:07,010 --> 00:39:09,460 I'm saying I don't have to worry about the rest of them 648 00:39:09,460 --> 00:39:12,790 because these are enough conditions to talk about 649 00:39:12,790 --> 00:39:14,180 everything I want to talk about. 650 00:39:14,180 --> 00:39:15,430 AUDIENCE: [UNINTELLIGIBLE]. 651 00:39:17,880 --> 00:39:19,350 PROFESSOR: I will define my measure as 652 00:39:19,350 --> 00:39:21,360 all of these things. 653 00:39:21,360 --> 00:39:23,540 Unfortunately, you need a little bit more, and if you 654 00:39:23,540 --> 00:39:26,750 want to get more you better take a course in real 655 00:39:26,750 --> 00:39:31,640 variables and measure theory. 656 00:39:31,640 --> 00:39:32,890 Good. 657 00:39:38,730 --> 00:39:46,960 So, if I want to now make this epsilon smaller, what I'm 658 00:39:46,960 --> 00:39:48,980 going to do is do it in a particular way. 659 00:39:48,980 --> 00:39:51,920 I'm going to start out partitioning this into 660 00:39:51,920 --> 00:39:54,270 intervals of size epsilon. 661 00:39:54,270 --> 00:39:57,800 Then I'm going to partition it into intervals of size 662 00:39:57,800 --> 00:39:59,700 epsilon over 2. 663 00:39:59,700 --> 00:40:04,150 When I partition it into intervals of size epsilon over 664 00:40:04,150 --> 00:40:07,970 2, I'm adding a bunch of extra things. 665 00:40:07,970 --> 00:40:13,580 This thing gets added because when I'm looking at the 666 00:40:13,580 --> 00:40:20,380 interval between epsilon and 3 epsilon over 2, the function 667 00:40:20,380 --> 00:40:29,100 is in this interval here, over this [UNINTELLIGIBLE]. 668 00:40:32,110 --> 00:40:37,990 It's in this interval over this whole thing. 669 00:40:37,990 --> 00:40:42,840 I'm representing it by this value down here. 670 00:40:42,840 --> 00:40:48,030 Now when I have this tinier interval, I see that this 671 00:40:48,030 --> 00:40:52,530 function is really in this interval from here to there 672 00:40:52,530 --> 00:40:57,050 also, and therefore, instead of representing the function 673 00:40:57,050 --> 00:41:01,210 over this interval by epsilon, I'm representing it by 3 674 00:41:01,210 --> 00:41:03,060 epsilon over 2. 675 00:41:03,060 --> 00:41:07,540 In other words, as I add these extra quantization levels, I 676 00:41:07,540 --> 00:41:11,850 can never lose anything, I only gain things. 677 00:41:11,850 --> 00:41:15,370 So I gain all of these cross-hatched regions when I 678 00:41:15,370 --> 00:41:20,260 do this, which says that when I add up all these things in 679 00:41:20,260 --> 00:41:25,960 the integral, every time I decrease epsilon by 2, the 680 00:41:25,960 --> 00:41:28,520 integral that I've got, the approximation to 681 00:41:28,520 --> 00:41:30,550 the integral increases. 682 00:41:30,550 --> 00:41:35,960 Now what happens when you take a sum of a set of numbers 683 00:41:35,960 --> 00:41:38,160 which are increasing? 684 00:41:38,160 --> 00:41:40,870 Well, they're increasing, the result that you get when you 685 00:41:40,870 --> 00:41:46,200 add them all up is increasing also, and therefore, as I go 686 00:41:46,200 --> 00:41:50,090 from epsilon to epsilon over 2 to epsilon over 4 and so 687 00:41:50,090 --> 00:41:53,130 forth, I keep climbing up. 688 00:41:53,130 --> 00:41:56,540 Conclusion, I either get to a finite number or I get to 689 00:41:56,540 --> 00:42:00,030 infinity -- only two possibilities. 690 00:42:00,030 --> 00:42:04,820 Which says that if I'm looking at non-negative functions, if 691 00:42:04,820 --> 00:42:08,570 I'm only looking at real functions which have 692 00:42:08,570 --> 00:42:13,070 non-negative values, the Lebesgue integral for a 693 00:42:13,070 --> 00:42:18,310 measurable function always exists, if I include infinite 694 00:42:18,310 --> 00:42:20,210 limits as well as finite limits. 695 00:42:23,860 --> 00:42:27,600 Now, if you think back to what you learned about integration, 696 00:42:27,600 --> 00:42:30,980 and I hope you at least learned enough about it that 697 00:42:30,980 --> 00:42:34,390 you remember there are a lot of very nasty conditions about 698 00:42:34,390 --> 00:42:38,950 when integrals exist and when they don't exist. 699 00:42:38,950 --> 00:42:40,640 Here that's all gone away. 700 00:42:43,230 --> 00:42:46,080 This is a beautifully simple statement. 701 00:42:46,080 --> 00:42:50,040 You take the integral of a non-negative function, if it's 702 00:42:50,040 --> 00:42:54,140 measurable, and there are only two possibilities. 703 00:42:54,140 --> 00:42:57,110 The integral of some finite number or 704 00:42:57,110 --> 00:42:59,340 the integral is infinite. 705 00:42:59,340 --> 00:43:01,950 It's never undefined, it's always defined. 706 00:43:06,010 --> 00:43:08,480 I think that's neat. 707 00:43:08,480 --> 00:43:12,680 A few people don't think it's neat, too bad. 708 00:43:17,100 --> 00:43:19,760 I guess when I was first studying this, I didn't think 709 00:43:19,760 --> 00:43:22,660 it was neat either because it was too complicated. 710 00:43:22,660 --> 00:43:26,450 So you have an excuse. 711 00:43:26,450 --> 00:43:29,030 If you think about it for a while and you understand it 712 00:43:29,030 --> 00:43:31,040 and you don't think it's neat, then I think 713 00:43:31,040 --> 00:43:32,290 you have a real problem. 714 00:43:36,090 --> 00:43:37,570 So now -- 715 00:43:40,590 --> 00:43:41,840 I did this. 716 00:43:45,926 --> 00:43:48,070 I'm getting too many slides out here. 717 00:43:51,970 --> 00:43:53,520 Here we go. 718 00:43:53,520 --> 00:43:55,860 Here's something new. 719 00:43:55,860 --> 00:43:57,790 Hardly looks new. 720 00:43:57,790 --> 00:44:02,220 Let's look at a function now, just defined on the 721 00:44:02,220 --> 00:44:05,430 interval zero to 1. 722 00:44:05,430 --> 00:44:11,420 Suppose that h of t is equal to 1 for each rational number 723 00:44:11,420 --> 00:44:16,510 and at zero for each irrational number. 724 00:44:16,510 --> 00:44:22,150 In other words, this is a function which looks 725 00:44:22,150 --> 00:44:23,640 absolutely wild. 726 00:44:26,190 --> 00:44:35,100 It just goes up to here and it's 1 or zero. 727 00:44:35,100 --> 00:44:39,590 It's 1 at this dense set of points, which we've already 728 00:44:39,590 --> 00:44:42,460 said doesn't amount to anything, and it's zero 729 00:44:42,460 --> 00:44:45,790 everywhere else. 730 00:44:45,790 --> 00:44:49,160 Now you put that into the Reimann integral, and the 731 00:44:49,160 --> 00:44:53,080 Reimann integral goes crazy, because no matter how small I 732 00:44:53,080 --> 00:44:56,820 make this interval, there are an infinite number of rational 733 00:44:56,820 --> 00:45:00,040 numbers in that interval, and therefore, the Reimann 734 00:45:00,040 --> 00:45:03,090 integral can never even get started. 735 00:45:03,090 --> 00:45:06,780 For the Lebesgue integral, on the other hand, look at what 736 00:45:06,780 --> 00:45:07,940 happens now. 737 00:45:07,940 --> 00:45:12,040 We have a bunch of points which are sitting at 1, we 738 00:45:12,040 --> 00:45:14,840 have a bunch of points which are sitting at zero. 739 00:45:14,840 --> 00:45:18,750 The only thing we have to do is evaluate the measure of the 740 00:45:18,750 --> 00:45:23,090 set of points which are up in some tiny interval up in here. 741 00:45:26,000 --> 00:45:30,360 What's this measure of the set of t's corresponding to the 742 00:45:30,360 --> 00:45:32,450 rational numbers? 743 00:45:32,450 --> 00:45:36,350 Well, you already said that was zero. 744 00:45:36,350 --> 00:45:40,490 Now, that's why Lebesgue integration works. 745 00:45:40,490 --> 00:45:45,260 Any countable set, and in fact, any of these uncountable 746 00:45:45,260 --> 00:45:51,110 sets that measure zero get lost in here because you're 747 00:45:51,110 --> 00:45:54,610 combining them all together and you say they don't 748 00:45:54,610 --> 00:45:58,490 contribute to the integral at all. 749 00:45:58,490 --> 00:46:01,660 That's why Lebesgue integration is so simple. 750 00:46:01,660 --> 00:46:03,530 You can forget about all that stuff. 751 00:46:08,410 --> 00:46:13,830 When we looked last time at the Fourier series for a 752 00:46:13,830 --> 00:46:18,960 square wave, you remember we found that everything behaved 753 00:46:18,960 --> 00:46:22,690 very nicely, except where the square wave had a 754 00:46:22,690 --> 00:46:26,350 discontinuity, the Fourier series converged to the 755 00:46:26,350 --> 00:46:31,430 mid-point, and that was kind of awkward. 756 00:46:31,430 --> 00:46:36,550 Well, the mid-points where the function is discontinuous 757 00:46:36,550 --> 00:46:40,530 don't amount to anything, because in that case there 758 00:46:40,530 --> 00:46:43,160 were just two of them, there were only two points. 759 00:46:43,160 --> 00:46:47,870 If they had measure zero it just washes away. 760 00:46:47,870 --> 00:46:51,030 You all felt intuitively when you saw that example, that 761 00:46:51,030 --> 00:46:53,690 those points were not important. 762 00:46:53,690 --> 00:46:57,370 You felt that this was mathematical carping. 763 00:46:57,370 --> 00:47:00,600 Well, Lebesgue felt it was mathematical carping too, but 764 00:47:00,600 --> 00:47:02,900 he went one step further and he said here's a way of 765 00:47:02,900 --> 00:47:04,710 getting rid of all of that and not having to 766 00:47:04,710 --> 00:47:06,540 worry about it anymore. 767 00:47:06,540 --> 00:47:09,250 So you've now gotten to the point where you don't have to 768 00:47:09,250 --> 00:47:14,660 worry about any of this stuff anymore. 769 00:47:20,920 --> 00:47:23,990 Now let's go a little bit further. 770 00:47:23,990 --> 00:47:26,730 We're almost at the end of this. 771 00:47:26,730 --> 00:47:31,640 If I take a function which maps the real numbers into the 772 00:47:31,640 --> 00:47:34,620 real numbers, in other words, it's a function which you can 773 00:47:34,620 --> 00:47:37,690 draw on the line. 774 00:47:37,690 --> 00:47:41,120 You take time going from minus infinity to plus infinity, you 775 00:47:41,120 --> 00:47:44,310 define what this function is at each time. 776 00:47:44,310 --> 00:47:46,910 That's what I'm talking about here, a function 777 00:47:46,910 --> 00:47:48,160 which you can draw. 778 00:47:51,150 --> 00:47:56,540 The functions magnitude of u of t, and the function 779 00:47:56,540 --> 00:48:03,910 magnitude of u of t squared are both non-negative. 780 00:48:03,910 --> 00:48:07,620 Now I'm not going to prove this, but it turns out that 781 00:48:07,620 --> 00:48:10,730 the magnitude and the magnitude squared are both 782 00:48:10,730 --> 00:48:14,350 measurable functions if u of t is a measurable function. 783 00:48:14,350 --> 00:48:17,190 In fact, from now on we're just going to assume that 784 00:48:17,190 --> 00:48:21,270 everything we deal with is measurable, every function is 785 00:48:21,270 --> 00:48:22,760 measurable. 786 00:48:22,760 --> 00:48:25,890 I challenge any of you without looking it up in a book to 787 00:48:25,890 --> 00:48:28,570 find an example of a non-measurable function. 788 00:48:28,570 --> 00:48:30,800 I challenge any of you to find an example of a 789 00:48:30,800 --> 00:48:33,830 non-measurable set. 790 00:48:33,830 --> 00:48:37,470 I challenge any of you to understand the definition of a 791 00:48:37,470 --> 00:48:40,590 non-measurable set if you look it up in a book. 792 00:48:40,590 --> 00:48:43,310 You've heard about things like the axiom of choice and things 793 00:48:43,310 --> 00:48:46,300 like that, which are very fishy kinds of things -- 794 00:48:46,300 --> 00:48:49,330 that's all involved in finding non-measurable function. 795 00:48:49,330 --> 00:48:53,150 So, any function that you think about is going to be 796 00:48:53,150 --> 00:48:56,300 measurable. 797 00:48:56,300 --> 00:49:00,080 I hate people who say things like that, but it's the only 798 00:49:00,080 --> 00:49:02,250 way to get around this because I don't want to give you any 799 00:49:02,250 --> 00:49:06,780 examples of that because they're awful. 800 00:49:06,780 --> 00:49:12,890 Since magnitude of u of t and magnitude of u of t squared 801 00:49:12,890 --> 00:49:16,220 are measurable and they're non-negative, 802 00:49:16,220 --> 00:49:18,750 their integrals exist. 803 00:49:18,750 --> 00:49:21,780 Their integrals exist and are either a finite number or 804 00:49:21,780 --> 00:49:24,040 they're infinite. 805 00:49:24,040 --> 00:49:29,710 So, we define L1 functions, and we'll be dealing with L1 806 00:49:29,710 --> 00:49:33,910 functions and L2 functions all the way through the course, u 807 00:49:33,910 --> 00:49:38,820 of t is an L1 function if it's measurable, and if this 808 00:49:38,820 --> 00:49:41,760 integrals is less than infinity. 809 00:49:41,760 --> 00:49:43,010 That's all there is to it. 810 00:49:45,440 --> 00:49:51,560 u2 is L2 if it's measurable and the integral of u of t 811 00:49:51,560 --> 00:49:55,290 squared is less than infinity. 812 00:49:55,290 --> 00:49:59,120 I could have said that at the beginning, but now you see 813 00:49:59,120 --> 00:50:03,040 that it makes a lot more sense than it did before because we 814 00:50:03,040 --> 00:50:06,810 know that if u of t is measurable, this integral 815 00:50:06,810 --> 00:50:10,990 exists -- it's either a finite number or infinity. 816 00:50:10,990 --> 00:50:14,920 The L1 functions are those particular functions where 817 00:50:14,920 --> 00:50:17,370 it's finite and not infinite. 818 00:50:17,370 --> 00:50:18,630 Same thing here. 819 00:50:18,630 --> 00:50:23,350 The L2 functions are those where this is finite. 820 00:50:23,350 --> 00:50:27,320 This is really the energy of the function, but now we can 821 00:50:27,320 --> 00:50:31,960 measure the energy of even weird functions which are zero 822 00:50:31,960 --> 00:50:37,250 on the irrationals and one on the rationals. 823 00:50:37,250 --> 00:50:41,840 Even things which are zero on the non-cantor set points and 824 00:50:41,840 --> 00:50:46,710 1 on the cantor set points, still it all works. 825 00:50:46,710 --> 00:50:53,850 So those define the set L1 and L2. 826 00:50:53,850 --> 00:50:59,560 Now, a complex function u of t, which maps r into c, why 827 00:50:59,560 --> 00:51:03,820 does a complex function map r into c? 828 00:51:03,820 --> 00:51:05,180 What's the r doing there? 829 00:51:10,470 --> 00:51:15,180 Think of any old complex function you can think of. 830 00:51:15,180 --> 00:51:18,155 e to the i, 2 pi t. 831 00:51:18,155 --> 00:51:22,410 That's something that wiggles around, the sinusoid. 832 00:51:22,410 --> 00:51:25,010 t is a real number. 833 00:51:25,010 --> 00:51:36,510 So that function e to the i 2 pi t is mapping real numbers 834 00:51:36,510 --> 00:51:39,120 into complex numbers. 835 00:51:39,120 --> 00:51:41,910 That's what we mean by something which maps the real 836 00:51:41,910 --> 00:51:44,130 numbers into complex numbers. 837 00:51:44,130 --> 00:51:46,480 We always call these complex functions. 838 00:51:49,120 --> 00:51:51,430 I mean mathematicians would say yeah, a 839 00:51:51,430 --> 00:51:53,470 function could be anything. 840 00:51:53,470 --> 00:51:56,700 But you know, when most of us think of a function, we're 841 00:51:56,700 --> 00:52:01,060 thinking of mapping a real variable into something else, 842 00:52:01,060 --> 00:52:03,230 and when we're thinking of mapping a real variable into 843 00:52:03,230 --> 00:52:06,970 something else, we're usually thinking of mapping it into 844 00:52:06,970 --> 00:52:10,610 real numbers or mapping it into complex numbers, and 845 00:52:10,610 --> 00:52:13,950 because we want to deal with these complex sinusoids, we 846 00:52:13,950 --> 00:52:16,320 have to include complex numbers also. 847 00:52:19,170 --> 00:52:23,620 So a complex function is measurable by definition if 848 00:52:23,620 --> 00:52:27,170 the real part and the imaginary part are each 849 00:52:27,170 --> 00:52:28,480 measurable. 850 00:52:28,480 --> 00:52:33,300 We already know when a real function is measurable. 851 00:52:33,300 --> 00:52:35,820 Namely, a real function is measurable if each of these 852 00:52:35,820 --> 00:52:37,550 slices are measurable. 853 00:52:37,550 --> 00:52:41,470 So now we know when a complex function is measurable. 854 00:52:41,470 --> 00:52:44,490 We already said that all of the complex functions you can 855 00:52:44,490 --> 00:52:47,480 think of and all the ones we'll ever deal with are all 856 00:52:47,480 --> 00:52:48,970 measurable. 857 00:52:48,970 --> 00:52:52,540 So L1 and L2 are defined in the same way when we're 858 00:52:52,540 --> 00:52:55,020 dealing with complex functions. 859 00:52:55,020 --> 00:52:58,390 Namely, just whether this integral is less than infinity 860 00:52:58,390 --> 00:53:01,230 and this integral is less than infinity. 861 00:53:03,730 --> 00:53:09,410 Since these functions -- this is a real function from real 862 00:53:09,410 --> 00:53:12,660 into real, this is a real function from real into real. 863 00:53:12,660 --> 00:53:14,740 So those are well-defined. 864 00:53:24,170 --> 00:53:26,930 What's the relationship between L1 865 00:53:26,930 --> 00:53:30,160 functions and L2 functions? 866 00:53:30,160 --> 00:53:34,080 Can a function be L1 and not L2? 867 00:53:34,080 --> 00:53:35,570 Can it be L2 and not L1? 868 00:53:39,090 --> 00:53:43,930 Yeah, it can be both, unfortunately. 869 00:53:43,930 --> 00:53:45,680 All possibilities exist. 870 00:53:45,680 --> 00:53:48,870 You can have functions which are neither L1 nor L2, 871 00:53:48,870 --> 00:53:52,100 functions that are L1 but not L2, functions that are L2 but 872 00:53:52,100 --> 00:53:55,940 not L1, and functions that are both L1 and L2. 873 00:53:55,940 --> 00:53:58,250 Those are the truly nice functions that we 874 00:53:58,250 --> 00:53:59,850 like to deal with. 875 00:53:59,850 --> 00:54:04,810 But there's one nice that you can say, and that follows from 876 00:54:04,810 --> 00:54:07,450 a simple argument here. 877 00:54:07,450 --> 00:54:10,370 If u of t is less than or equal to 1, if the magnitude 878 00:54:10,370 --> 00:54:12,600 of u of t is less than or equal to 1--. 879 00:54:19,390 --> 00:54:22,660 Let's start out by looking at u of t being greater than or 880 00:54:22,660 --> 00:54:23,430 equal to 1. 881 00:54:23,430 --> 00:54:26,950 If u of t is greater than or equal to 1, then u squared of 882 00:54:26,950 --> 00:54:29,830 t is even bigger. 883 00:54:29,830 --> 00:54:32,250 You see the thing that happened is when u of t 884 00:54:32,250 --> 00:54:34,980 becomes bigger than t, u squared of t 885 00:54:34,980 --> 00:54:36,920 becomes even more bigger. 886 00:54:36,920 --> 00:54:41,420 When u of t is less than 1, u squared of t is 887 00:54:41,420 --> 00:54:42,760 less than u of t. 888 00:54:46,960 --> 00:54:51,330 But if u of t is less than or equal to 1, it's less than 1. 889 00:54:51,330 --> 00:54:56,290 So in all cases, for all t, u of t, magnitude is less than 890 00:54:56,290 --> 00:55:02,070 or equal to u of t squared plus 1. 891 00:55:02,070 --> 00:55:04,090 So that takes into account both cases. 892 00:55:04,090 --> 00:55:06,540 It's a bound. 893 00:55:06,540 --> 00:55:13,500 So if I'm looking at functions which only exist between over 894 00:55:13,500 --> 00:55:18,280 some limited time interval, and I take the integral from 895 00:55:18,280 --> 00:55:22,480 minus t over 2 to the t over 2 of the magnitude of u of t, I 896 00:55:22,480 --> 00:55:25,910 get something which is less than or equal to the integral 897 00:55:25,910 --> 00:55:31,000 of u squared of t plus the integral of 1, and the 898 00:55:31,000 --> 00:55:35,740 integral of 1 over this finite limit is just t. 899 00:55:35,740 --> 00:55:44,870 This says that if a function is L2 and the function only 900 00:55:44,870 --> 00:55:51,110 exists over a finite interval, then the function is L1 also. 901 00:55:51,110 --> 00:55:54,490 So as long as I'm dealing with Fourier series, as long as I'm 902 00:55:54,490 --> 00:56:00,270 dealing with finite duration functions, L2 means L1. 903 00:56:00,270 --> 00:56:03,550 All of the nice things that you get with L1 functions 904 00:56:03,550 --> 00:56:07,760 apply to L2 functions also, and there are a lot of nice 905 00:56:07,760 --> 00:56:09,920 things that happen for L1 functions. 906 00:56:09,920 --> 00:56:14,340 There are a lot of nice things that happen for L2 functions. 907 00:56:14,340 --> 00:56:18,310 You take the union of what happens for L1 and for L2, and 908 00:56:18,310 --> 00:56:20,740 that's beautiful. 909 00:56:20,740 --> 00:56:23,460 You can say anything then. 910 00:56:23,460 --> 00:56:27,395 Can't calculate anything, of course, but we all said, we 911 00:56:27,395 --> 00:56:30,480 leave that to computers. 912 00:56:30,480 --> 00:56:32,530 Let's go back Fourier series now, let's go 913 00:56:32,530 --> 00:56:34,300 back to the real world. 914 00:56:37,890 --> 00:56:44,450 Any old function we have u of t, the magnitude of u of t and 915 00:56:44,450 --> 00:56:49,402 the magnitude of u of t times either the 2 pi ift, for any 916 00:56:49,402 --> 00:56:54,490 old f, this thing has magnitude 1, right, a complex 917 00:56:54,490 --> 00:56:57,060 exponential. 918 00:56:57,060 --> 00:56:59,620 Real f, real t. 919 00:56:59,620 --> 00:57:02,400 This just has magnitude 1. 920 00:57:02,400 --> 00:57:07,360 And therefore, this magnitude is equal to this magnitude. 921 00:57:07,360 --> 00:57:12,660 This says that if the function u of t is L1, then the 922 00:57:12,660 --> 00:57:21,070 function u of t times either the 2 pi IFT is also L1, which 923 00:57:21,070 --> 00:57:23,710 says if we can integrate one we can integrate the other. 924 00:57:27,060 --> 00:57:31,200 In other words, the integral of u of t, either the 2 pi 925 00:57:31,200 --> 00:57:35,190 ift, the magnitude of the fdt is going to 926 00:57:35,190 --> 00:57:38,750 be less than infinity. 927 00:57:38,750 --> 00:57:40,960 Since we're taking the magnitude, it's either finite 928 00:57:40,960 --> 00:57:45,520 or it's infinite, and since u of t in magnitude when we 929 00:57:45,520 --> 00:57:48,630 integrate it is less than infinity, this thing is less 930 00:57:48,630 --> 00:57:50,750 than infinity also. 931 00:57:50,750 --> 00:57:54,230 Now, this is a complex number in here. 932 00:57:54,230 --> 00:57:58,260 So you can break it up into a real part and an imaginary 933 00:57:58,260 --> 00:58:04,350 part, and if this whole thing, if the magnitude is less than 934 00:58:04,350 --> 00:58:13,250 infinity, then the magnitude of the real part is finite. 935 00:58:13,250 --> 00:58:17,430 If you take the real part over the region where this is 936 00:58:17,430 --> 00:58:22,110 positive and the region where it's negative, you still get 937 00:58:22,110 --> 00:58:23,460 non-negative numbers. 938 00:58:23,460 --> 00:58:26,160 In other words, if we're taking the integral of 939 00:58:26,160 --> 00:58:28,660 something which has positive values -- 940 00:58:28,660 --> 00:58:33,080 I should have written this out in more detail, it's not 941 00:58:33,080 --> 00:58:34,330 enough to--. 942 00:58:41,790 --> 00:58:44,010 I'm taking the integral of something which--. 943 00:58:51,040 --> 00:58:52,290 This is u of t. 944 00:58:58,740 --> 00:59:00,120 If I can find another color. 945 00:59:07,340 --> 00:59:10,990 Let me draw magnitude of u of t on top of this. 946 00:59:16,510 --> 00:59:22,690 This thing here is magnitude of u of t. 947 00:59:22,690 --> 00:59:23,640 What? 948 00:59:23,640 --> 00:59:25,670 AUDIENCE: [INAUDIBLE]. 949 00:59:25,670 --> 00:59:27,390 PROFESSOR: I'm making it real for the time being because I'm 950 00:59:27,390 --> 00:59:28,660 just looking at the real part. 951 00:59:32,480 --> 00:59:34,840 In other words, what I'm looking is this quantity here. 952 00:59:37,610 --> 00:59:41,820 Later you can imagine doing the same thing out on complex 953 00:59:41,820 --> 00:59:43,070 numbers, OK? 954 00:59:45,660 --> 00:59:48,000 So what I'm saying is if I just look at the real part of 955 00:59:48,000 --> 00:59:51,390 u of t -- call this real part of u of t, if you like. 956 00:59:56,250 --> 01:00:03,390 If I know that this is finite, this is non-negative, I know 957 01:00:03,390 --> 01:00:07,670 that the positive part of this function has a finite 958 01:00:07,670 --> 01:00:11,300 integral, I know that the negative part of it has a 959 01:00:11,300 --> 01:00:13,270 finite integral. 960 01:00:13,270 --> 01:00:17,170 In other words, the thing which makes integration messy 961 01:00:17,170 --> 01:00:21,760 is you sometimes have the positive part being infinite, 962 01:00:21,760 --> 01:00:24,860 the negative part being infinite also, and the two of 963 01:00:24,860 --> 01:00:27,180 them cancel out somehow to give you something finite. 964 01:00:29,680 --> 01:00:32,290 When you're dealing with the magnitude of u of t, if the 965 01:00:32,290 --> 01:00:35,960 magnitude of u of t has an infinite integral, then that 966 01:00:35,960 --> 01:00:37,880 messy thing can't happen. 967 01:00:37,880 --> 01:00:42,850 It says that the positive part has a finite integral, the 968 01:00:42,850 --> 01:00:45,510 negative part has a finite integral also. 969 01:00:45,510 --> 01:00:49,160 If you take the imaginary part, visualize that out this 970 01:00:49,160 --> 01:00:53,660 way, the positive part of the imaginary part has a finite 971 01:00:53,660 --> 01:00:57,430 integral, the negative part of the imaginary part has a 972 01:00:57,430 --> 01:00:59,610 finite integral also. 973 01:00:59,610 --> 01:01:07,380 It says if the magnitude of u of t, that integral always 974 01:01:07,380 --> 01:01:11,920 exists, and if it's finite, then these positive parts of 975 01:01:11,920 --> 01:01:15,690 the real, the positive part of the imaginary part, all of 976 01:01:15,690 --> 01:01:21,150 those are finite, and it says the integral itself is finite. 977 01:01:21,150 --> 01:01:26,130 Which says that this integral here has to be finite. 978 01:01:26,130 --> 01:01:29,880 Namely, the positive part of the real part, the negative 979 01:01:29,880 --> 01:01:33,460 part of the real part, the positive part of the imaginary 980 01:01:33,460 --> 01:01:37,800 part, the negative part of the imaginary part, all four have 981 01:01:37,800 --> 01:01:44,840 to be finite, just because this quantity here is finite. 982 01:01:44,840 --> 01:01:50,130 Now, if u of 2 is L2 and also time limited, it's L1, and the 983 01:01:50,130 --> 01:01:53,020 same conclusion follows. 984 01:01:53,020 --> 01:01:56,130 So this integral always exists if it's 985 01:01:56,130 --> 01:01:57,380 over a finite interval. 986 01:02:01,780 --> 01:02:05,090 So at this point we're really ready to go back to this 987 01:02:05,090 --> 01:02:09,580 theorem about Fourier series that we stated last time and 988 01:02:09,580 --> 01:02:13,290 which was a little bit mysterious at that point. 989 01:02:13,290 --> 01:02:16,920 In fact, at this point we've already proven part of it. 990 01:02:16,920 --> 01:02:20,650 I wanted to prove it because I wanted you to know that not 991 01:02:20,650 --> 01:02:23,460 everything in measure theory is difficult. 992 01:02:23,460 --> 01:02:27,040 An awful lot of these things, after you know just a very 993 01:02:27,040 --> 01:02:31,880 small number of the ideas, are very, very simple. 994 01:02:31,880 --> 01:02:34,770 Now, you will think this is not simple because you haven't 995 01:02:34,770 --> 01:02:36,895 had time to think about the ten slides 996 01:02:36,895 --> 01:02:39,620 that have gone before. 997 01:02:39,620 --> 01:02:41,920 But if you go back and you look at them again, if you 998 01:02:41,920 --> 01:02:44,630 read the notes, you will see that, in fact, it all is 999 01:02:44,630 --> 01:02:46,860 pretty simple. 1000 01:02:46,860 --> 01:02:51,760 What this says is if u of t, a complex function real into c, 1001 01:02:51,760 --> 01:02:58,820 but time limited, suppose it's an L2 function, then it's also 1002 01:02:58,820 --> 01:03:04,610 L1 over that interval minus t over 2 to plus t over 2. 1003 01:03:04,610 --> 01:03:10,320 Then for each k and z, then for each integer k, this 1004 01:03:10,320 --> 01:03:16,930 integral here, this function here, is now an L1 function, 1005 01:03:16,930 --> 01:03:21,340 and therefore, this integral exists, is finite. 1006 01:03:21,340 --> 01:03:23,890 You divide by t is still finite. 1007 01:03:23,890 --> 01:03:28,660 So that Fourier coefficient has to exist and it has to 1008 01:03:28,660 --> 01:03:30,200 exist as a finite value. 1009 01:03:32,870 --> 01:03:36,220 Now, you look at Reimann integration, and you look at 1010 01:03:36,220 --> 01:03:39,410 the theorems about Reimann integration, and if they're 1011 01:03:39,410 --> 01:03:43,280 stated by somebody who was stating theorems, the 1012 01:03:43,280 --> 01:03:47,150 conditions are monstrous. 1013 01:03:47,150 --> 01:03:49,490 This is not monstrous. 1014 01:03:49,490 --> 01:03:53,870 It says all you need is measurability and L1, which 1015 01:03:53,870 --> 01:03:56,290 says it doesn't go up to infinity. 1016 01:03:56,290 --> 01:03:58,490 That's enough to say that every one of these Fourier 1017 01:03:58,490 --> 01:04:00,830 coefficients has to exist. 1018 01:04:04,830 --> 01:04:07,310 It might be hard to integrate it, but you 1019 01:04:07,310 --> 01:04:08,960 know it has to exist. 1020 01:04:08,960 --> 01:04:13,180 Now the next thing it says is that -- this is more 1021 01:04:13,180 --> 01:04:14,880 complicated. 1022 01:04:14,880 --> 01:04:18,870 What we would like to say and what we tried to say before 1023 01:04:18,870 --> 01:04:22,170 and what you like to say with the Fourier series is that u 1024 01:04:22,170 --> 01:04:28,270 of t is equal to this, where you sum from minus infinity to 1025 01:04:28,270 --> 01:04:30,220 plus infinity. 1026 01:04:30,220 --> 01:04:32,420 We saw that we can't say that. 1027 01:04:32,420 --> 01:04:37,670 We saw that we can't it for functions which have step 1028 01:04:37,670 --> 01:04:41,120 discontinuities, because whenever you have a step 1029 01:04:41,120 --> 01:04:44,480 discontinuity, the Fourier series converges to the 1030 01:04:44,480 --> 01:04:47,110 mid-point of that discontinuity. 1031 01:04:47,110 --> 01:04:51,200 If you were unfortunate enough to try to make life simple and 1032 01:04:51,200 --> 01:04:58,120 define the function without defining it at the step 1033 01:04:58,120 --> 01:05:02,000 discontinuity as the mid-point, then the Fourier 1034 01:05:02,000 --> 01:05:04,510 series would not be equal to u of t. 1035 01:05:04,510 --> 01:05:07,860 But what this says is that if you take the difference 1036 01:05:07,860 --> 01:05:14,500 between u of t and a finite expansion, and then you look 1037 01:05:14,500 --> 01:05:18,550 at the energy in that difference, it says that the 1038 01:05:18,550 --> 01:05:20,670 energy and the difference goes to zero. 1039 01:05:23,930 --> 01:05:29,020 Now that's far more important than having this integral be 1040 01:05:29,020 --> 01:05:34,990 equal to that, because frankly, we don't care a fig 1041 01:05:34,990 --> 01:05:41,010 for whether this is equal to this at every t or not. 1042 01:05:41,010 --> 01:05:45,330 What we care about is when we add more and more terms onto 1043 01:05:45,330 --> 01:05:47,050 this Fourier series. 1044 01:05:47,050 --> 01:05:50,280 I mean in engineering we're always approximating things. 1045 01:05:50,280 --> 01:05:53,010 We have to approximate things. 1046 01:05:53,010 --> 01:05:56,600 We talk about functions u of t, but our functions u of t 1047 01:05:56,600 --> 01:06:00,160 are just models of things anyway, and we want those 1048 01:06:00,160 --> 01:06:04,450 models to really converge to something, which means that as 1049 01:06:04,450 --> 01:06:07,670 we take more and more terms in the Fourier series, we get 1050 01:06:07,670 --> 01:06:11,110 something which comes closer to u of t and it comes closer 1051 01:06:11,110 --> 01:06:12,310 in energy terms. 1052 01:06:12,310 --> 01:06:16,090 Remember when we take this, when we approximate it in this 1053 01:06:16,090 --> 01:06:19,940 way by a finite Fourier series, and we then quantize 1054 01:06:19,940 --> 01:06:29,160 coefficients, and then we go back to the function, we have 1055 01:06:29,160 --> 01:06:33,580 lost all the coefficients and all of these terms for the 1056 01:06:33,580 --> 01:06:34,840 very high frequencies. 1057 01:06:34,840 --> 01:06:37,090 We've dropped them off. 1058 01:06:37,090 --> 01:06:42,460 What this says is if we take more and more of them and then 1059 01:06:42,460 --> 01:06:46,490 we quantize and we go back to a function v of t it says as 1060 01:06:46,490 --> 01:06:50,080 we add more and more Fourier coefficients and quantize 1061 01:06:50,080 --> 01:06:53,585 carefully, we can come closer and closer to the function 1062 01:06:53,585 --> 01:06:56,680 that we started with. 1063 01:06:56,680 --> 01:07:01,410 You don't get that by talking about point to point equality, 1064 01:07:01,410 --> 01:07:05,600 because as soon as we quantize you lose all of that anyway. 1065 01:07:05,600 --> 01:07:08,640 You don't have a quality anywhere anymore, and the only 1066 01:07:08,640 --> 01:07:12,560 thing you can hope for is a small mean square error. 1067 01:07:15,370 --> 01:07:19,230 So, this looks more complicated than what you 1068 01:07:19,230 --> 01:07:23,360 would like, but what I'm trying to tell you is that 1069 01:07:23,360 --> 01:07:26,650 this is far more important than what you would like. 1070 01:07:26,650 --> 01:07:29,190 What you would like is not important at all. 1071 01:07:29,190 --> 01:07:33,030 I can give you examples of things where this converges to 1072 01:07:33,030 --> 01:07:37,270 this everywhere, but in fact, no matter how many terms you 1073 01:07:37,270 --> 01:07:41,390 take, the energy difference between these two things is 1074 01:07:41,390 --> 01:07:43,480 very large. 1075 01:07:43,480 --> 01:07:45,610 Those are the ugly things. 1076 01:07:45,610 --> 01:07:52,560 That says you never get a good approximation, even though 1077 01:07:52,560 --> 01:07:56,980 looking at things point-wise everything looks great. 1078 01:07:56,980 --> 01:07:59,380 But what you're really interested in is these mean 1079 01:07:59,380 --> 01:08:04,710 square approximations and what this Fourier series thing says 1080 01:08:04,710 --> 01:08:08,790 is that if you deal with measurable functions then this 1081 01:08:08,790 --> 01:08:12,960 converges just to that in this very nice energy sense, and 1082 01:08:12,960 --> 01:08:15,580 energy is what we're interested in. 1083 01:08:15,580 --> 01:08:16,700 The final part of this -- 1084 01:08:16,700 --> 01:08:20,590 I mean I talked about this a little bit last time -- 1085 01:08:20,590 --> 01:08:24,020 sometimes instead of starting out with a function and 1086 01:08:24,020 --> 01:08:27,010 approximating it with the Fourier series, you want to 1087 01:08:27,010 --> 01:08:29,140 start out with the coefficients 1088 01:08:29,140 --> 01:08:30,830 and find the function. 1089 01:08:30,830 --> 01:08:34,140 In fact, when we start talking about modulation, that's 1090 01:08:34,140 --> 01:08:37,270 exactly what we're going to be doing because we're always 1091 01:08:37,270 --> 01:08:40,250 going to be starting out with these digital sequences, we're 1092 01:08:40,250 --> 01:08:43,790 going to be finding functions from the digital sequences. 1093 01:08:43,790 --> 01:08:46,810 This final part of the theorem says yes, you can 1094 01:08:46,810 --> 01:08:48,380 do that and it works. 1095 01:08:48,380 --> 01:08:53,730 It says that given any set of coefficients where the 1096 01:08:53,730 --> 01:08:56,550 coefficients have finite energy, in other words, where 1097 01:08:56,550 --> 01:09:02,070 the sum is less than infinity, then there's an L2 function, u 1098 01:09:02,070 --> 01:09:08,470 of t, which satisfies the above in this limiting sense. 1099 01:09:08,470 --> 01:09:10,680 Now this is the hardest thing of the theorem to prove. 1100 01:09:10,680 --> 01:09:13,840 It looks obvious but it's not. 1101 01:09:13,840 --> 01:09:15,680 But anyway, it's there. 1102 01:09:15,680 --> 01:09:20,790 It says these approximations are rock solid and you can now 1103 01:09:20,790 --> 01:09:24,820 forget about all of this measure theoretic stuff, and 1104 01:09:24,820 --> 01:09:28,220 you can just live with the fact that all of these in 1105 01:09:28,220 --> 01:09:31,340 terms of L2 approximations, work. 1106 01:09:39,030 --> 01:09:41,510 I mean again, it doesn't look beautiful at this point 1107 01:09:41,510 --> 01:09:44,380 because you haven't seen it for long enough. 1108 01:09:44,380 --> 01:09:45,370 It really is beautiful. 1109 01:09:45,370 --> 01:09:47,150 It's beautiful stuff. 1110 01:09:47,150 --> 01:09:52,050 When you think about it long enough, for 40 years is how 1111 01:09:52,050 --> 01:09:55,140 long I've been thinking about it, it becomes more and more 1112 01:09:55,140 --> 01:09:56,370 beautiful every year. 1113 01:09:56,370 --> 01:09:59,740 So, if you live long enough it'll be beautiful. 1114 01:10:04,120 --> 01:10:07,950 Any time you talk about this type of convergence, we call 1115 01:10:07,950 --> 01:10:15,040 it convergence in the mean, and then notation we will use 1116 01:10:15,040 --> 01:10:20,180 is l.i.m., which looks like limit but which really stands 1117 01:10:20,180 --> 01:10:24,140 for limit in the mean. 1118 01:10:24,140 --> 01:10:28,070 We will write that complicated thing on the last slide, which 1119 01:10:28,070 --> 01:10:31,780 I said is not really that complicated, but we will write 1120 01:10:31,780 --> 01:10:35,930 this as this. 1121 01:10:35,930 --> 01:10:39,290 In other words, when we write limit in the mean, we mean 1122 01:10:39,290 --> 01:10:43,250 that the difference between these two sides in energy 1123 01:10:43,250 --> 01:10:46,360 sense goes to zero as k gets large. 1124 01:10:46,360 --> 01:10:48,360 That's what this limit means. 1125 01:10:48,360 --> 01:10:51,650 It just means the statement that we talked about before. 1126 01:10:51,650 --> 01:10:57,420 So the Fourier series, the theorem really says you can 1127 01:10:57,420 --> 01:11:01,080 find all of the Fourier coefficients in this way, they 1128 01:11:01,080 --> 01:11:05,610 all exist, they all exist exactly, they're all finite. 1129 01:11:05,610 --> 01:11:10,530 The function then exists as this limit in the mean, which 1130 01:11:10,530 --> 01:11:13,250 is the thing that we're always interested in. 1131 01:11:13,250 --> 01:11:19,760 So, every L2 function defined over a finite interval, and 1132 01:11:19,760 --> 01:11:23,700 every L1 function defined over a finite interval, all of 1133 01:11:23,700 --> 01:11:27,420 these have a Fourier series which satisfies these two 1134 01:11:27,420 --> 01:11:30,930 relationships. 1135 01:11:30,930 --> 01:11:33,000 Now, what we're going to do with all of this is we're 1136 01:11:33,000 --> 01:11:38,090 going to segment an arbitrary function, an arbitrary L2 1137 01:11:38,090 --> 01:11:44,460 function over the entire time range, and we're going to 1138 01:11:44,460 --> 01:11:50,760 segment it into pieces, all of width t. 1139 01:11:50,760 --> 01:11:54,530 Then we're going to expand each of those segments into a 1140 01:11:54,530 --> 01:11:55,780 Fourier series. 1141 01:11:58,480 --> 01:12:03,890 That's what people do any time they compress voice that I 1142 01:12:03,890 --> 01:12:04,960 said several times. 1143 01:12:04,960 --> 01:12:07,060 When you compress voice -- 1144 01:12:07,060 --> 01:12:09,810 for some reason or other everybody does it in 20 1145 01:12:09,810 --> 01:12:11,960 millisecond increments. 1146 01:12:11,960 --> 01:12:15,930 You chop up the voice into 20 millisecond increments. 1147 01:12:15,930 --> 01:12:19,090 You then, at least, conceptually look at those 20 1148 01:12:19,090 --> 01:12:22,625 millisecond increments, think of them in terms of the 1149 01:12:22,625 --> 01:12:25,930 Fourier series, you expand them in the Fourier series. 1150 01:12:25,930 --> 01:12:29,400 So for each increment in time we get a different Fourier 1151 01:12:29,400 --> 01:12:32,590 series, and we use those Fourier series to approximate 1152 01:12:32,590 --> 01:12:34,270 the function. 1153 01:12:34,270 --> 01:12:38,990 So, each one of the increments now is going to be represented 1154 01:12:38,990 --> 01:12:41,170 in this way here. 1155 01:12:41,170 --> 01:12:45,330 Since we're only looking at the function now over the 1156 01:12:45,330 --> 01:12:51,710 interval around time, mt, we look at it this way, we 1157 01:12:51,710 --> 01:12:55,540 calculate the coefficients, again, in terms of this 1158 01:12:55,540 --> 01:13:00,980 rectangular function spaced off by m. 1159 01:13:00,980 --> 01:13:05,170 This is saying the same thing that we said before, just 1160 01:13:05,170 --> 01:13:12,530 moving it from minus t over 2 to t over 2 to t minus mt. 1161 01:13:21,990 --> 01:13:24,730 Here's minus t over 2. 1162 01:13:24,730 --> 01:13:27,560 The t over 2. 1163 01:13:27,560 --> 01:13:32,050 Next we're looking at t over 2 to 3t over 2. 1164 01:13:32,050 --> 01:13:36,280 Next we're looking at 5t over 2. 1165 01:13:36,280 --> 01:13:39,690 This is m equals zero. 1166 01:13:39,690 --> 01:13:41,990 This is m equals 1. 1167 01:13:41,990 --> 01:13:44,950 This is m equals 2. 1168 01:13:44,950 --> 01:13:49,350 This notation here you should get used to it, it will be 1169 01:13:49,350 --> 01:13:51,570 confusing for a while. 1170 01:13:51,570 --> 01:13:58,720 All it means is that, and it works. 1171 01:13:58,720 --> 01:14:01,350 So, in fact, you can take all these terms, you can break 1172 01:14:01,350 --> 01:14:02,600 them up this way. 1173 01:14:05,120 --> 01:14:08,070 So what you've really done is you've broken u of t into a 1174 01:14:08,070 --> 01:14:12,180 double sub expansion. 1175 01:14:12,180 --> 01:14:19,150 These exponentials limited in time, that entire set of 1176 01:14:19,150 --> 01:14:22,320 functions are talking to each other. 1177 01:14:22,320 --> 01:14:25,020 A function living in this interval of time and a 1178 01:14:25,020 --> 01:14:27,790 function living in this interval of time have to be 1179 01:14:27,790 --> 01:14:31,250 orthogonal, because I multiply this by this and I get 1180 01:14:31,250 --> 01:14:34,570 zero at each time. 1181 01:14:34,570 --> 01:14:39,000 In one interval these exponentials are orthogonal to 1182 01:14:39,000 --> 01:14:40,940 each other -- we've pointed that out before. 1183 01:14:40,940 --> 01:14:46,830 So we have a doubly exponential, we have a double 1184 01:14:46,830 --> 01:14:51,450 sum of orthogonal functions, and what we're saying is that 1185 01:14:51,450 --> 01:14:56,290 any old L2 function at all we can break up into 1186 01:14:56,290 --> 01:14:59,940 this kind of sum here. 1187 01:14:59,940 --> 01:15:05,090 This is a very complicated way of saying what we think of 1188 01:15:05,090 --> 01:15:06,740 physically anyway. 1189 01:15:06,740 --> 01:15:10,020 It says take a big long function, segment it into 1190 01:15:10,020 --> 01:15:14,390 intervals of length t, break up each interval of length t 1191 01:15:14,390 --> 01:15:15,680 into a Fourier series. 1192 01:15:15,680 --> 01:15:18,300 That's all that's saying. 1193 01:15:18,300 --> 01:15:20,580 So it's saying what is obvious. 1194 01:15:20,580 --> 01:15:24,500 We're going to find a number of such orthogonal expansions 1195 01:15:24,500 --> 01:15:27,200 which work for arbitrary L2 functions. 1196 01:15:29,920 --> 01:15:32,700 As I said, it's a conceptual basis for voice compress 1197 01:15:32,700 --> 01:15:34,770 algorithms. 1198 01:15:34,770 --> 01:15:38,480 Even more, next time we're going to go 1199 01:15:38,480 --> 01:15:40,610 into the Fourier integral. 1200 01:15:40,610 --> 01:15:43,760 You think of the Fourier integral as being the right 1201 01:15:43,760 --> 01:15:50,290 way to go from time to frequency, but, in fact, it's 1202 01:15:50,290 --> 01:15:54,670 not really the right way to go from time to frequency. 1203 01:15:54,670 --> 01:16:00,300 When we think of voice or you think of the wave form of a 1204 01:16:00,300 --> 01:16:06,310 symphony, for example, what going on there? 1205 01:16:06,310 --> 01:16:09,780 Over every little interval of time you hear various 1206 01:16:09,780 --> 01:16:12,770 frequencies, right? 1207 01:16:12,770 --> 01:16:19,260 In fact, if you make t equal to the timing of the music, 1208 01:16:19,260 --> 01:16:24,220 the idea becomes very, very clean because at each time 1209 01:16:24,220 --> 01:16:26,090 somebody changes a note. 1210 01:16:26,090 --> 01:16:29,590 So you go from one frequency to another frequency, so it's 1211 01:16:29,590 --> 01:16:32,290 a rather clean way of looking at this. 1212 01:16:32,290 --> 01:16:36,860 Our notion of frequency that we think of intuitively is 1213 01:16:36,860 --> 01:16:42,500 much more closely associated with the idea of frequencies 1214 01:16:42,500 --> 01:16:45,620 is changing in time then it is of frequencies 1215 01:16:45,620 --> 01:16:46,880 being constant in time. 1216 01:16:46,880 --> 01:16:52,050 When we look at a Fourier integral, everything 1217 01:16:52,050 --> 01:16:53,380 is frozen in time. 1218 01:16:53,380 --> 01:16:55,910 As soon as you take the Fourier integral, you've 1219 01:16:55,910 --> 01:16:59,520 glopped everything from time minus infinity to time plus 1220 01:16:59,520 --> 01:17:01,610 infinity all together. 1221 01:17:01,610 --> 01:17:04,410 Here when we look at these expansions, we're looking at 1222 01:17:04,410 --> 01:17:06,880 these frequencies changing. 1223 01:17:06,880 --> 01:17:10,510 There's an unfortunate part about frequencies changing, 1224 01:17:10,510 --> 01:17:13,410 and that is frequencies, unfortunately, live over the 1225 01:17:13,410 --> 01:17:14,910 entire time interval. 1226 01:17:14,910 --> 01:17:18,540 These truncated frequencies work very nicely but they 1227 01:17:18,540 --> 01:17:22,590 don't quite correspond to non-truncated time intervals, 1228 01:17:22,590 --> 01:17:26,030 but it still does match our intuition and this is a useful 1229 01:17:26,030 --> 01:17:29,770 way to think about functions that change in time. 1230 01:17:34,310 --> 01:17:37,840 If you believe what I've just said, why do people ever worry 1231 01:17:37,840 --> 01:17:39,660 about the Fourier integral at all? 1232 01:17:42,650 --> 01:17:46,830 Well, you see the problem is this quantity t here that I'm 1233 01:17:46,830 --> 01:17:52,350 segmenting over is unnatural. 1234 01:17:52,350 --> 01:17:55,990 If I look at voice, there's no t that you can take which is a 1235 01:17:55,990 --> 01:17:56,720 natural quantity. 1236 01:17:56,720 --> 01:18:00,330 The 20 milliseconds is just something arbitrary that 1237 01:18:00,330 --> 01:18:03,360 somebody did once and it worked. 1238 01:18:03,360 --> 01:18:06,780 Too many other engineers in the field said, well that 1239 01:18:06,780 --> 01:18:09,420 works, I'll pick the same thing instead of trying to 1240 01:18:09,420 --> 01:18:12,450 think through what a better number would be. 1241 01:18:12,450 --> 01:18:14,210 So they all use the same t. 1242 01:18:14,210 --> 01:18:16,880 It doesn't correspond to anything physically. 1243 01:18:16,880 --> 01:18:20,310 So, in fact, people do try the same thing. 1244 01:18:20,310 --> 01:18:24,100 Let me just say what a Fourier transform is and we'll talk 1245 01:18:24,100 --> 01:18:27,150 about it most of next time because that's the next thing 1246 01:18:27,150 --> 01:18:28,260 we have to deal with. 1247 01:18:28,260 --> 01:18:31,510 Something you're all familiar with are these Fourier 1248 01:18:31,510 --> 01:18:33,900 transform relationships. 1249 01:18:33,900 --> 01:18:36,290 There's the function of time, there's 1250 01:18:36,290 --> 01:18:38,010 the function of frequency. 1251 01:18:38,010 --> 01:18:43,000 You can go from one to the other, mapping. 1252 01:18:43,000 --> 01:18:45,640 Well, if you know this you can find this. 1253 01:18:45,640 --> 01:18:47,930 If you know this, you can find this. 1254 01:18:47,930 --> 01:18:50,580 It looks a little bit like the Fourier series, and usually 1255 01:18:50,580 --> 01:18:53,530 when people learn about the Fourier integral they start 1256 01:18:53,530 --> 01:18:58,050 with a Fourier series and start letting t become large, 1257 01:18:58,050 --> 01:19:00,700 which doesn't quite work. 1258 01:19:00,700 --> 01:19:04,730 But anyway, that's what we're going to do. 1259 01:19:04,730 --> 01:19:07,180 If the function is well-behaved, the first 1260 01:19:07,180 --> 01:19:10,260 integral exists and the second exists. 1261 01:19:10,260 --> 01:19:12,560 What does well-behaved mean? 1262 01:19:12,560 --> 01:19:14,340 It means what we usually mean by 1263 01:19:14,340 --> 01:19:18,050 well-behaved, it means it works. 1264 01:19:18,050 --> 01:19:23,640 So again, this theorem here is another example, or not, as 1265 01:19:23,640 --> 01:19:25,330 the case may be. 1266 01:19:25,330 --> 01:19:28,710 But we'll make this clearer next time.