WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.920 align:middle line:90% [RUSTLING] 00:00:03.920 --> 00:00:06.860 align:middle line:90% [CLICKING] 00:00:06.860 --> 00:00:18.630 align:middle line:90% 00:00:18.630 --> 00:00:21.660 align:middle line:84% CASEY RODRIGUEZ: OK, let's continue our discussion 00:00:21.660 --> 00:00:22.620 align:middle line:90% of measurable sets. 00:00:22.620 --> 00:00:25.160 align:middle line:90% 00:00:25.160 --> 00:00:30.500 align:middle line:84% Let me just briefly recall for you, the end of last time, 00:00:30.500 --> 00:00:33.850 align:middle line:84% we discussed some general notions, 00:00:33.850 --> 00:00:40.350 align:middle line:84% some special collections of subsets of R, 00:00:40.350 --> 00:00:42.240 align:middle line:84% one of those being an algebra, which 00:00:42.240 --> 00:00:44.490 align:middle line:84% is closed under taking complements and taking 00:00:44.490 --> 00:00:45.750 align:middle line:90% finite unions. 00:00:45.750 --> 00:00:49.050 align:middle line:84% And then we said a collection of subsets of R's is a sigma 00:00:49.050 --> 00:00:55.290 align:middle line:84% algebra if it's also closed under taking countable unions. 00:00:55.290 --> 00:00:59.040 align:middle line:84% And not every algebra is a sigma algebra. 00:00:59.040 --> 00:01:00.720 align:middle line:90% Why did we bring all this up? 00:01:00.720 --> 00:01:05.740 align:middle line:84% Well, we were in the middle of discussing Lebesgue measurable 00:01:05.740 --> 00:01:06.240 align:middle line:90% sets. 00:01:06.240 --> 00:01:17.830 align:middle line:84% So recall that we say E is Lebesgue measurable 00:01:17.830 --> 00:01:26.860 align:middle line:84% if for all subsets of R A, the outer measure of A 00:01:26.860 --> 00:01:32.050 align:middle line:84% is equal to the outer measure of A intersect E 00:01:32.050 --> 00:01:37.240 align:middle line:84% plus the outer measure of A intersect the complement. 00:01:37.240 --> 00:01:40.690 align:middle line:90% 00:01:40.690 --> 00:01:42.970 align:middle line:84% So in some sense, a measurable set 00:01:42.970 --> 00:01:48.130 align:middle line:84% is one that divides sets nicely with respect to outer measure. 00:01:48.130 --> 00:01:51.520 align:middle line:90% 00:01:51.520 --> 00:02:02.230 align:middle line:84% And we denoted script M to be the set of all E such 00:02:02.230 --> 00:02:07.930 align:middle line:90% that E is Lebesgue measurable. 00:02:07.930 --> 00:02:10.300 align:middle line:84% And I'm going to stop saying Lebesgue measurable 00:02:10.300 --> 00:02:12.175 align:middle line:84% and just say measurable from now on. 00:02:12.175 --> 00:02:15.380 align:middle line:90% 00:02:15.380 --> 00:02:18.320 align:middle line:84% And what we showed last time, we showed 00:02:18.320 --> 00:02:22.580 align:middle line:84% that M, the set of measurable subsets of R, 00:02:22.580 --> 00:02:24.060 align:middle line:90% do form an algebra. 00:02:24.060 --> 00:02:26.930 align:middle line:84% So it follows from the definition 00:02:26.930 --> 00:02:30.290 align:middle line:84% that if E is measurable, then E complement is measurable. 00:02:30.290 --> 00:02:32.510 align:middle line:84% But we also showed that if I take 00:02:32.510 --> 00:02:36.640 align:middle line:84% a finite union of measurable sets, 00:02:36.640 --> 00:02:39.050 align:middle line:84% or a finite collection of measurable sets, 00:02:39.050 --> 00:02:42.092 align:middle line:90% their union is measurable. 00:02:42.092 --> 00:02:43.550 align:middle line:84% And what we're going to do today is 00:02:43.550 --> 00:02:47.600 align:middle line:84% we're going to show that the set, the collection 00:02:47.600 --> 00:02:50.120 align:middle line:84% of measurable, Lebesgue measurable, subsets of R 00:02:50.120 --> 00:02:51.510 align:middle line:90% form a sigma algebra. 00:02:51.510 --> 00:02:55.730 align:middle line:84% So they have this stronger property. 00:02:55.730 --> 00:03:03.955 align:middle line:90% And also that-- last time-- 00:03:03.955 --> 00:03:07.500 align:middle line:90% 00:03:07.500 --> 00:03:08.910 align:middle line:84% there's no way to write this in-- 00:03:08.910 --> 00:03:12.920 align:middle line:90% 00:03:12.920 --> 00:03:17.370 align:middle line:84% maybe not definition, but notation-- 00:03:17.370 --> 00:03:29.120 align:middle line:84% that B is the Borel sigma algebra, which is, recall, 00:03:29.120 --> 00:03:30.780 align:middle line:84% we proved it at the end of last time. 00:03:30.780 --> 00:03:34.640 align:middle line:84% So we gave this as an example of a sigma algebra. 00:03:34.640 --> 00:03:43.230 align:middle line:84% This is the smallest sigma algebra 00:03:43.230 --> 00:03:50.170 align:middle line:90% containing all open sets. 00:03:50.170 --> 00:03:54.420 align:middle line:84% So any other sigma algebra that contains all open sets 00:03:54.420 --> 00:03:58.380 align:middle line:84% contains the sigma algebra B, the Borel sigma algebra. 00:03:58.380 --> 00:04:07.130 align:middle line:90% 00:04:07.130 --> 00:04:11.240 align:middle line:84% So like I said, our goal for this lecture 00:04:11.240 --> 00:04:15.140 align:middle line:84% is we're going to show that this collection of Lebesgue 00:04:15.140 --> 00:04:17.390 align:middle line:84% measurable sets is a sigma algebra. 00:04:17.390 --> 00:04:19.950 align:middle line:84% We're going to show it contains the Borel sigma algebra. 00:04:19.950 --> 00:04:24.350 align:middle line:90% And then what is-- 00:04:24.350 --> 00:04:30.110 align:middle line:84% I mean, I can even state it now, or at least 00:04:30.110 --> 00:04:31.610 align:middle line:90% I'll just say in words. 00:04:31.610 --> 00:04:32.930 align:middle line:90% What is Lebesgue measure? 00:04:32.930 --> 00:04:36.350 align:middle line:84% It's simply-- Lebesgue measure of a measurable set 00:04:36.350 --> 00:04:42.925 align:middle line:84% will be the outer measure of that set as long as it's 00:04:42.925 --> 00:04:44.890 align:middle line:90% Lebesgue measurable. 00:04:44.890 --> 00:04:47.260 align:middle line:84% So what we're going to do is we're 00:04:47.260 --> 00:04:50.155 align:middle line:84% going to show that M is a sigma algebra. 00:04:50.155 --> 00:04:53.000 align:middle line:90% 00:04:53.000 --> 00:05:09.120 align:middle line:84% Now, one preliminary remark that or lemma 00:05:09.120 --> 00:05:11.940 align:middle line:84% that we're going to prove is that so this condition 3 00:05:11.940 --> 00:05:16.320 align:middle line:84% says that you need to prove that every countable collection is 00:05:16.320 --> 00:05:20.580 align:middle line:84% closed under taking unions, to ensure that an algebra is 00:05:20.580 --> 00:05:21.760 align:middle line:90% a sigma algebra. 00:05:21.760 --> 00:05:23.460 align:middle line:84% But in fact, you don't have to check it 00:05:23.460 --> 00:05:26.568 align:middle line:90% for an arbitrary collection. 00:05:26.568 --> 00:05:28.110 align:middle line:84% You really just need to check it only 00:05:28.110 --> 00:05:31.890 align:middle line:84% for a countable collection of disjoint subsets. 00:05:31.890 --> 00:05:38.330 align:middle line:84% So that's the lemma that we'll use 00:05:38.330 --> 00:05:42.050 align:middle line:84% when we prove that the collection of Lebesgue 00:05:42.050 --> 00:05:46.520 align:middle line:84% measurable sets is a sigma algebra is the following-- 00:05:46.520 --> 00:05:47.450 align:middle line:90% so this is general. 00:05:47.450 --> 00:05:49.340 align:middle line:84% This has nothing to do, necessarily, 00:05:49.340 --> 00:05:51.200 align:middle line:90% with Lebesgue measure. 00:05:51.200 --> 00:06:04.010 align:middle line:90% Let A be an algebra in-- 00:06:04.010 --> 00:06:05.950 align:middle line:84% so again, N here is a natural number-- 00:06:05.950 --> 00:06:10.720 align:middle line:90% 00:06:10.720 --> 00:06:18.460 align:middle line:84% a collection of elements of A. So each of these 00:06:18.460 --> 00:06:30.150 align:middle line:84% is a subset of R. Then there exists a countable collection F 00:06:30.150 --> 00:06:54.250 align:middle line:84% n of elements of A that are disjoint-- 00:06:54.250 --> 00:06:57.310 align:middle line:90% 00:06:57.310 --> 00:07:02.700 align:middle line:84% so if I take F n and F m, and n does not equal m, 00:07:02.700 --> 00:07:04.890 align:middle line:84% then their intersection is empty-- 00:07:04.890 --> 00:07:17.190 align:middle line:84% such that the union of the E n's equals the union of the F n's. 00:07:17.190 --> 00:07:18.400 align:middle line:90% So what does this say? 00:07:18.400 --> 00:07:19.610 align:middle line:90% This says if we have-- 00:07:19.610 --> 00:07:23.190 align:middle line:90% 00:07:23.190 --> 00:07:27.250 align:middle line:90% let me make this a remark, then. 00:07:27.250 --> 00:07:29.695 align:middle line:90% What is the conclusion of this? 00:07:29.695 --> 00:07:37.970 align:middle line:90% 00:07:37.970 --> 00:07:41.690 align:middle line:84% We only need to check this third condition 00:07:41.690 --> 00:07:46.130 align:middle line:84% for being a sigma algebra, that a countable collection is 00:07:46.130 --> 00:07:52.520 align:middle line:84% closed under taking unions, condition 3 for sigma algebra, 00:07:52.520 --> 00:08:08.710 align:middle line:84% for disjoint collections of elements. 00:08:08.710 --> 00:08:12.163 align:middle line:90% 00:08:12.163 --> 00:08:13.830 align:middle line:84% This follows immediately from this lemma 00:08:13.830 --> 00:08:18.120 align:middle line:84% because if I have any arbitrary collection of elements 00:08:18.120 --> 00:08:23.400 align:middle line:84% of the algebra, then their union is equal to a union of elements 00:08:23.400 --> 00:08:28.800 align:middle line:84% of A, where the union is now over elements that 00:08:28.800 --> 00:08:32.320 align:middle line:90% are disjoint from each other. 00:08:32.320 --> 00:08:34.360 align:middle line:90% So this is the point. 00:08:34.360 --> 00:08:37.419 align:middle line:84% I only need to check the condition for an algebra 00:08:37.419 --> 00:08:40.120 align:middle line:84% to be a sigma algebra by checking 00:08:40.120 --> 00:08:43.720 align:middle line:84% that countable unions of disjoint sets 00:08:43.720 --> 00:08:44.800 align:middle line:90% remain in the algebra. 00:08:44.800 --> 00:08:51.800 align:middle line:90% 00:08:51.800 --> 00:08:53.225 align:middle line:90% So the proof is not very hard. 00:08:53.225 --> 00:08:56.980 align:middle line:90% 00:08:56.980 --> 00:09:05.160 align:middle line:84% So let G-- let's call it something-- 00:09:05.160 --> 00:09:12.990 align:middle line:84% G n to be the union k equals 1 to n of E k. 00:09:12.990 --> 00:09:18.470 align:middle line:90% 00:09:18.470 --> 00:09:27.080 align:middle line:84% So then, these are growing because G n contains 00:09:27.080 --> 00:09:33.680 align:middle line:84% the first n E k's, and then n plus 1, I tack on another one. 00:09:33.680 --> 00:09:38.870 align:middle line:84% So G1 is contained in G2 is contained in G3. 00:09:38.870 --> 00:09:47.180 align:middle line:84% And this is just kind of easy to check. 00:09:47.180 --> 00:09:50.180 align:middle line:84% And I'll leave it to you that the union of the E n's is equal 00:09:50.180 --> 00:09:53.820 align:middle line:90% to the union of the G n's. 00:09:53.820 --> 00:09:56.270 align:middle line:84% I mean, the G n's are really just unions of the E 00:09:56.270 --> 00:09:58.460 align:middle line:90% k's, finite unions of the E k's. 00:09:58.460 --> 00:10:03.480 align:middle line:90% So this is pretty clear. 00:10:03.480 --> 00:10:08.040 align:middle line:90% 00:10:08.040 --> 00:10:11.190 align:middle line:84% I mean, if you actually want to sit down and do the argument, 00:10:11.190 --> 00:10:17.640 align:middle line:84% every element in G n is contained in E1 up to E n. 00:10:17.640 --> 00:10:20.520 align:middle line:84% And so every G n is contained in this union. 00:10:20.520 --> 00:10:22.260 align:middle line:84% And therefore, the union of the G n's is 00:10:22.260 --> 00:10:23.610 align:middle line:90% contained in this union. 00:10:23.610 --> 00:10:25.950 align:middle line:84% And then it's easy to go the other way 00:10:25.950 --> 00:10:28.498 align:middle line:84% around showing this union is contained in this union 00:10:28.498 --> 00:10:29.415 align:middle line:90% so that they're equal. 00:10:29.415 --> 00:10:32.780 align:middle line:90% 00:10:32.780 --> 00:10:47.280 align:middle line:84% And now, I take F1 to be G1 and F 00:10:47.280 --> 00:10:51.540 align:middle line:84% n plus 1 to be F n plus 1 take away 00:10:51.540 --> 00:10:56.430 align:middle line:84% F n for n bigger than or equal to 1, or I should say-- 00:10:56.430 --> 00:11:06.730 align:middle line:84% sorry about that, that should be G n plus 1 take away G n. 00:11:06.730 --> 00:11:09.130 align:middle line:90% So what is this? 00:11:09.130 --> 00:11:12.130 align:middle line:84% I take the first set to be G1 and then 00:11:12.130 --> 00:11:15.520 align:middle line:84% F2 will be what's in G2 but not in G1. 00:11:15.520 --> 00:11:22.795 align:middle line:90% So then, what is this? 00:11:22.795 --> 00:11:25.870 align:middle line:90% 00:11:25.870 --> 00:11:26.780 align:middle line:90% What do we get? 00:11:26.780 --> 00:11:29.980 align:middle line:84% We get that the union of any finite guy 00:11:29.980 --> 00:11:33.070 align:middle line:90% is equal to, in fact-- 00:11:33.070 --> 00:11:40.330 align:middle line:90% 00:11:40.330 --> 00:11:45.820 align:middle line:84% so this is pretty clear, again, because what am I doing here? 00:11:45.820 --> 00:11:49.180 align:middle line:84% F2 will be what's in G2 take away G1. 00:11:49.180 --> 00:11:53.560 align:middle line:84% What's in F3 is whatever's in G3 take away G2. 00:11:53.560 --> 00:11:58.360 align:middle line:84% And their union is going to be the same as over here 00:11:58.360 --> 00:12:02.230 align:middle line:84% because I'm including G1 and so on. 00:12:02.230 --> 00:12:03.550 align:middle line:90% I hope this is clear. 00:12:03.550 --> 00:12:07.980 align:middle line:90% 00:12:07.980 --> 00:12:10.170 align:middle line:84% And in fact, I don't know why I'm writing n. 00:12:10.170 --> 00:12:14.740 align:middle line:90% 00:12:14.740 --> 00:12:18.940 align:middle line:84% We in fact get this union is equal to this union. 00:12:18.940 --> 00:12:21.290 align:middle line:84% And again, I'm leaving some details out, 00:12:21.290 --> 00:12:25.000 align:middle line:84% but you can check that this union is contained 00:12:25.000 --> 00:12:26.530 align:middle line:84% in this union and then this union 00:12:26.530 --> 00:12:29.110 align:middle line:84% is contained in this union very easily. 00:12:29.110 --> 00:12:37.880 align:middle line:84% But I hope the idea is pretty clear because at each stage, 00:12:37.880 --> 00:12:41.090 align:middle line:84% you're taking whatever is in this set, 00:12:41.090 --> 00:12:45.920 align:middle line:84% cutting out what was appeared before it, 00:12:45.920 --> 00:12:47.000 align:middle line:90% in the sets before it. 00:12:47.000 --> 00:13:02.010 align:middle line:90% 00:13:02.010 --> 00:13:07.050 align:middle line:84% So now, let's go back to measurable sets. 00:13:07.050 --> 00:13:11.700 align:middle line:90% We're now going to show that-- 00:13:11.700 --> 00:13:17.180 align:middle line:84% almost-- we are going to prove that the collection of Lebesgue 00:13:17.180 --> 00:13:20.280 align:middle line:84% measurable sets is a sigma algebra. 00:13:20.280 --> 00:13:25.000 align:middle line:84% But first, we need the following theorem, 00:13:25.000 --> 00:13:26.070 align:middle line:90% which is the following-- 00:13:26.070 --> 00:13:28.910 align:middle line:90% 00:13:28.910 --> 00:13:34.270 align:middle line:84% Let A be a subset of R. And this is essentially the whole game, 00:13:34.270 --> 00:13:35.560 align:middle line:90% as we'll see. 00:13:35.560 --> 00:13:43.730 align:middle line:84% Let A be a subset of R E1 up to E n Lebesgue 00:13:43.730 --> 00:13:49.060 align:middle line:84% measurable, and not just measurable, 00:13:49.060 --> 00:14:03.160 align:middle line:90% but be disjoint measurable sets. 00:14:03.160 --> 00:14:04.930 align:middle line:84% So I only have finitely many of them. 00:14:04.930 --> 00:14:08.610 align:middle line:90% 00:14:08.610 --> 00:14:17.230 align:middle line:84% Then, the outer measure of A intersect 00:14:17.230 --> 00:14:28.730 align:middle line:84% the union of these guys is equal to the sum k equals 1 to n 00:14:28.730 --> 00:14:38.540 align:middle line:84% of the outer measure of A intersect E k. 00:14:38.540 --> 00:14:54.280 align:middle line:90% 00:14:54.280 --> 00:14:59.790 align:middle line:84% So this shouldn't come as too much of a surprise 00:14:59.790 --> 00:15:06.700 align:middle line:84% because this is kind of close to saying-- 00:15:06.700 --> 00:15:15.110 align:middle line:84% I should say this is very true if say E1-- 00:15:15.110 --> 00:15:16.820 align:middle line:90% say you just have two sets-- 00:15:16.820 --> 00:15:20.210 align:middle line:90% E1 is E and E2 is E complement. 00:15:20.210 --> 00:15:22.490 align:middle line:84% Then that's just the definition of being measurable. 00:15:22.490 --> 00:15:27.730 align:middle line:90% 00:15:27.730 --> 00:15:30.970 align:middle line:84% And you build up from that by using induction. 00:15:30.970 --> 00:15:33.370 align:middle line:90% That's the basic argument. 00:15:33.370 --> 00:15:35.860 align:middle line:84% Again, I'm sorry I keep moving away from the chalkboard 00:15:35.860 --> 00:15:39.160 align:middle line:84% because I'm used to lecturing to a classroom, 00:15:39.160 --> 00:15:43.600 align:middle line:84% even though it's been months now since I've been in a classroom 00:15:43.600 --> 00:15:45.670 align:middle line:90% with students. 00:15:45.670 --> 00:15:47.155 align:middle line:84% And because I'm a freak of nature 00:15:47.155 --> 00:15:49.030 align:middle line:84% who writes from my left hand, I have to write 00:15:49.030 --> 00:15:51.100 align:middle line:90% and then do this kind of move. 00:15:51.100 --> 00:15:54.760 align:middle line:84% But anyways, we're going to prove this by induction. 00:15:54.760 --> 00:16:02.500 align:middle line:90% 00:16:02.500 --> 00:16:04.530 align:middle line:90% So the proof is by induction. 00:16:04.530 --> 00:16:09.040 align:middle line:90% 00:16:09.040 --> 00:16:11.220 align:middle line:90% So n equals 1. 00:16:11.220 --> 00:16:12.510 align:middle line:90% That's just one set. 00:16:12.510 --> 00:16:13.275 align:middle line:90% This is clear. 00:16:13.275 --> 00:16:16.330 align:middle line:90% 00:16:16.330 --> 00:16:19.750 align:middle line:84% That's just the outer measure of A intersect E1 00:16:19.750 --> 00:16:22.720 align:middle line:84% is equal to the outer measure of A intersect E1. 00:16:22.720 --> 00:16:23.575 align:middle line:90% So that's fine. 00:16:23.575 --> 00:16:27.300 align:middle line:90% 00:16:27.300 --> 00:16:29.940 align:middle line:84% So I'm going to put a check there 00:16:29.940 --> 00:16:31.680 align:middle line:90% without writing anything down. 00:16:31.680 --> 00:16:38.680 align:middle line:84% So now, let's label this statement by star. 00:16:38.680 --> 00:16:40.545 align:middle line:84% So now, let's do the induction step. 00:16:40.545 --> 00:16:47.990 align:middle line:90% 00:16:47.990 --> 00:16:58.490 align:middle line:84% So suppose star holds for n equals m. 00:16:58.490 --> 00:17:03.320 align:middle line:84% And what we now want to show, meaning that if I have E1 up 00:17:03.320 --> 00:17:09.619 align:middle line:84% to up to E m disjoint measurable sets, then star holds. 00:17:09.619 --> 00:17:12.485 align:middle line:84% Now, I want to show that statement holds for m plus 1. 00:17:12.485 --> 00:17:28.500 align:middle line:90% 00:17:28.500 --> 00:17:47.510 align:middle line:84% So let E1 up to E m plus 1 be measurable disjoint sets. 00:17:47.510 --> 00:17:48.827 align:middle line:90% So each of them-- 00:17:48.827 --> 00:17:49.910 align:middle line:90% they're pairwise disjoint. 00:17:49.910 --> 00:17:58.260 align:middle line:90% 00:17:58.260 --> 00:18:15.260 align:middle line:84% Then, since E m plus 1 is measurable, what do we have? 00:18:15.260 --> 00:18:18.910 align:middle line:90% 00:18:18.910 --> 00:18:25.440 align:middle line:84% Well, first, before I use that, let me just note something. 00:18:25.440 --> 00:18:36.600 align:middle line:84% Since E m plus 1 is just disjoint from E1 up to E m, 00:18:36.600 --> 00:18:40.080 align:middle line:84% let A be a subset of R. We want to verify star for A, 00:18:40.080 --> 00:18:45.140 align:middle line:90% so I need to have an A. 00:18:45.140 --> 00:18:51.800 align:middle line:84% So since E k intersect E m plus 1 00:18:51.800 --> 00:18:58.150 align:middle line:84% equals the empty set for all k equals 1 to m, 00:18:58.150 --> 00:19:08.065 align:middle line:84% we get that A intersect the union k equals 1 m plus 1. 00:19:08.065 --> 00:19:11.000 align:middle line:90% 00:19:11.000 --> 00:19:12.320 align:middle line:90% What does this equal to? 00:19:12.320 --> 00:19:15.640 align:middle line:90% 00:19:15.640 --> 00:19:19.840 align:middle line:84% Or then I intersect that with E m plus 1, 00:19:19.840 --> 00:19:25.780 align:middle line:84% this is simply equal to A intersect E m plus 1 00:19:25.780 --> 00:19:30.560 align:middle line:84% because E1 up to E m-- these are disjoint. 00:19:30.560 --> 00:19:37.240 align:middle line:84% So when I transfer this through, I 00:19:37.240 --> 00:19:40.930 align:middle line:84% can write this intersection by bringing this intersect, 00:19:40.930 --> 00:19:44.590 align:middle line:84% each of these E k's, where k equals 1 to m plus 1. 00:19:44.590 --> 00:19:47.530 align:middle line:84% And then this is empty when k is not equal to m plus 1. 00:19:47.530 --> 00:19:53.360 align:middle line:84% So I just pick up E m plus 1 and I get that there. 00:19:53.360 --> 00:20:07.610 align:middle line:84% But also, if I look at intersect complement, what is this? 00:20:07.610 --> 00:20:09.080 align:middle line:90% This is equal to-- 00:20:09.080 --> 00:20:14.890 align:middle line:84% now k, E k for k going from 1 to m, 00:20:14.890 --> 00:20:19.810 align:middle line:84% these are all contained in the complement of E m plus 1 00:20:19.810 --> 00:20:21.290 align:middle line:90% because they're disjoint. 00:20:21.290 --> 00:20:28.570 align:middle line:84% So this is equal to just A intersect union 00:20:28.570 --> 00:20:34.030 align:middle line:90% A equals 1 to m E k. 00:20:34.030 --> 00:20:39.620 align:middle line:90% 00:20:39.620 --> 00:20:42.950 align:middle line:84% Now, I got ahead of myself a minute ago, 00:20:42.950 --> 00:20:44.900 align:middle line:90% but now we're at this stage. 00:20:44.900 --> 00:20:48.340 align:middle line:90% 00:20:48.340 --> 00:20:58.190 align:middle line:84% Since E m plus 1 is measurable, the measure 00:20:58.190 --> 00:21:08.570 align:middle line:84% of A intersect this union A equals 1 m plus 1 E k, this 00:21:08.570 --> 00:21:16.220 align:middle line:84% is equal to the outer measure of A intersect k 00:21:16.220 --> 00:21:23.840 align:middle line:84% equals 1 m plus 1 E k intersect E m 00:21:23.840 --> 00:21:33.390 align:middle line:84% plus 1 plus the outer measure of A intersect union k 00:21:33.390 --> 00:21:47.520 align:middle line:84% equals 1 m plus 1 E k intersect E m plus 1 complement. 00:21:47.520 --> 00:21:50.970 align:middle line:84% Just using that E n plus 1 is measurable. 00:21:50.970 --> 00:21:53.670 align:middle line:84% And now, we plug in what these things are. 00:21:53.670 --> 00:22:00.310 align:middle line:84% So this whole thing is equal to A intersect E m plus 1. 00:22:00.310 --> 00:22:08.290 align:middle line:84% So this is equal to the measure of A intersect E m plus 1. 00:22:08.290 --> 00:22:14.700 align:middle line:84% And now, this whole thing is equal to the measure 00:22:14.700 --> 00:22:18.795 align:middle line:84% of A intersect the union only going up to m. 00:22:18.795 --> 00:22:22.640 align:middle line:90% 00:22:22.640 --> 00:22:25.280 align:middle line:84% And now, this is where we use the induction hypothesis 00:22:25.280 --> 00:22:30.750 align:middle line:84% because now we have a union of m disjoint measurable sets. 00:22:30.750 --> 00:22:41.630 align:middle line:84% So this is equal to plus sum from k 00:22:41.630 --> 00:22:50.400 align:middle line:90% equals 1 to m A intersect E k. 00:22:50.400 --> 00:22:54.870 align:middle line:84% So this is by our induction hypothesis. 00:22:54.870 --> 00:22:57.710 align:middle line:90% 00:22:57.710 --> 00:23:01.880 align:middle line:84% And combining these two terms is exactly what we 00:23:01.880 --> 00:23:05.150 align:middle line:84% want for n equals m plus 1, k equals 00:23:05.150 --> 00:23:15.510 align:middle line:84% 1 m plus 1 outer measure of A intersect E k. 00:23:15.510 --> 00:23:16.590 align:middle line:90% And that's the proof. 00:23:16.590 --> 00:23:30.080 align:middle line:90% 00:23:30.080 --> 00:23:38.730 align:middle line:84% So now using this theorem and the lemma before it, 00:23:38.730 --> 00:23:40.695 align:middle line:84% we will prove that the collection 00:23:40.695 --> 00:23:43.395 align:middle line:84% m of Lebesgue measurable sets is a sigma algebra. 00:23:43.395 --> 00:23:48.120 align:middle line:90% 00:23:48.120 --> 00:23:50.320 align:middle line:84% We already know it's an algebra up to this point. 00:23:50.320 --> 00:23:52.350 align:middle line:84% But we just need to verify it's a sigma algebra. 00:23:52.350 --> 00:24:03.510 align:middle line:90% 00:24:03.510 --> 00:24:04.560 align:middle line:90% So what's the proof? 00:24:04.560 --> 00:24:11.840 align:middle line:84% Now, again, based on this remark here, 00:24:11.840 --> 00:24:16.535 align:middle line:84% we only need to verify that if I have 00:24:16.535 --> 00:24:22.970 align:middle line:84% a countable collection of disjoint measurable sets, then 00:24:22.970 --> 00:24:24.620 align:middle line:90% the union is measurable. 00:24:24.620 --> 00:24:27.260 align:middle line:84% I don't have to check it for every collection 00:24:27.260 --> 00:24:31.280 align:middle line:84% of measurable sets, just disjoint measurable sets. 00:24:31.280 --> 00:24:35.410 align:middle line:90% 00:24:35.410 --> 00:24:36.860 align:middle line:90% So we just need to check. 00:24:36.860 --> 00:24:53.080 align:middle line:84% So we've already checked it's an algebra. 00:24:53.080 --> 00:25:03.240 align:middle line:84% So by the lemma, the first thing we proved during this lecture, 00:25:03.240 --> 00:25:26.140 align:middle line:84% we just need to show m is closed under taking countable disjoint 00:25:26.140 --> 00:25:28.990 align:middle line:84% unions, meaning if I have a countable collection 00:25:28.990 --> 00:25:30.520 align:middle line:84% of disjoint measurable sets, I need 00:25:30.520 --> 00:25:32.540 align:middle line:90% to show the union is measurable. 00:25:32.540 --> 00:25:46.240 align:middle line:84% So let E sub n be a countable collection here-- 00:25:46.240 --> 00:25:47.770 align:middle line:90% n is a natural number-- 00:25:47.770 --> 00:25:54.030 align:middle line:90% of disjoint measurable sets. 00:25:54.030 --> 00:25:56.800 align:middle line:90% 00:25:56.800 --> 00:25:59.970 align:middle line:84% So we need to verify the definition of being measurable. 00:25:59.970 --> 00:26:05.910 align:middle line:84% But remember, that reduces really to one inequality, 00:26:05.910 --> 00:26:11.870 align:middle line:84% since one of those inequalities is always clear. 00:26:11.870 --> 00:26:21.960 align:middle line:84% So let A be a subset of R, and let's denote E to be the union. 00:26:21.960 --> 00:26:25.210 align:middle line:90% 00:26:25.210 --> 00:26:26.410 align:middle line:90% So what do we need to show? 00:26:26.410 --> 00:26:31.780 align:middle line:90% 00:26:31.780 --> 00:26:37.380 align:middle line:84% We want to show that the outer measure of A intersect E 00:26:37.380 --> 00:26:43.200 align:middle line:84% complement plus the outer measure of A intersect E-- 00:26:43.200 --> 00:26:45.990 align:middle line:90% E here, again, is a union-- 00:26:45.990 --> 00:26:48.420 align:middle line:84% is equal to the outer measure of A. 00:26:48.420 --> 00:26:51.750 align:middle line:84% But we always have the outer measure of A 00:26:51.750 --> 00:26:54.495 align:middle line:84% is less than or equal to this, so we just need to show this. 00:26:54.495 --> 00:27:21.070 align:middle line:90% 00:27:21.070 --> 00:27:23.710 align:middle line:90% So let's do this. 00:27:23.710 --> 00:27:28.550 align:middle line:90% 00:27:28.550 --> 00:27:29.990 align:middle line:90% Let N be a natural number. 00:27:29.990 --> 00:27:37.630 align:middle line:84% So let N be a natural number by-- 00:27:37.630 --> 00:27:51.510 align:middle line:84% so since M is an algebra, this, a finite union, 00:27:51.510 --> 00:27:58.250 align:middle line:84% n equals 1 to capital N of E sub n is measurable. 00:27:58.250 --> 00:28:06.640 align:middle line:84% And therefore, if I want the outer measure of A, 00:28:06.640 --> 00:28:12.040 align:middle line:84% this is equal to the outer measure 00:28:12.040 --> 00:28:18.940 align:middle line:84% of A intersect n equals 1 to N E sub 00:28:18.940 --> 00:28:27.640 align:middle line:84% n plus the outer measure of A intersect n 00:28:27.640 --> 00:28:34.120 align:middle line:90% equals 1 to N E n complement. 00:28:34.120 --> 00:28:40.450 align:middle line:84% Now, this finite union is contained 00:28:40.450 --> 00:28:44.750 align:middle line:90% in E. E is the total union. 00:28:44.750 --> 00:28:49.130 align:middle line:84% And therefore, the complement of this finite union 00:28:49.130 --> 00:28:53.190 align:middle line:90% contains the complement of E. 00:28:53.190 --> 00:28:58.620 align:middle line:84% So this complement here contains the complement of E, 00:28:58.620 --> 00:29:02.250 align:middle line:84% and therefore, A intersect this is a bigger 00:29:02.250 --> 00:29:04.950 align:middle line:84% set than A intersect the complement of E. 00:29:04.950 --> 00:29:09.200 align:middle line:84% So this is bigger than or equal to the outer measure 00:29:09.200 --> 00:29:19.070 align:middle line:84% of A intersect union n equals 1 to N E n plus the outer measure 00:29:19.070 --> 00:29:22.490 align:middle line:84% of A intersect E complement-- again, 00:29:22.490 --> 00:29:26.060 align:middle line:84% because this finite union is contained in the whole union. 00:29:26.060 --> 00:29:28.910 align:middle line:84% So when I take complements, that switches around 00:29:28.910 --> 00:29:30.110 align:middle line:90% what's contained in what. 00:29:30.110 --> 00:29:32.540 align:middle line:84% So then I get A intersect E complement 00:29:32.540 --> 00:29:34.790 align:middle line:84% is contained in A intersect the complement 00:29:34.790 --> 00:29:35.690 align:middle line:90% of this finite union. 00:29:35.690 --> 00:29:41.293 align:middle line:90% 00:29:41.293 --> 00:29:41.960 align:middle line:90% And that's good. 00:29:41.960 --> 00:29:43.460 align:middle line:90% So this is now up here. 00:29:43.460 --> 00:29:47.420 align:middle line:84% We wanted this to be smaller than the measure of A here. 00:29:47.420 --> 00:29:50.010 align:middle line:90% And now what do we have? 00:29:50.010 --> 00:29:56.270 align:middle line:84% We have the outer measure of A with a finite union 00:29:56.270 --> 00:29:58.160 align:middle line:90% of disjoint measurable sets. 00:29:58.160 --> 00:30:07.300 align:middle line:84% And we can write this using the previous theorem as sum from n 00:30:07.300 --> 00:30:12.360 align:middle line:84% equals 1 to capital N outer measure 00:30:12.360 --> 00:30:16.515 align:middle line:90% of A intersect E n plus-- 00:30:16.515 --> 00:30:20.910 align:middle line:90% 00:30:20.910 --> 00:30:24.030 align:middle line:90% just keeping the second term. 00:30:24.030 --> 00:30:27.450 align:middle line:84% Now, this holds for every N. N was arbitrary. 00:30:27.450 --> 00:30:32.380 align:middle line:90% 00:30:32.380 --> 00:30:34.480 align:middle line:84% So I can let capital N go to infinity. 00:30:34.480 --> 00:30:37.840 align:middle line:84% Remember, I've shown that the outer measure of A 00:30:37.840 --> 00:30:40.790 align:middle line:84% is bigger than or equal to this quantity here. 00:30:40.790 --> 00:30:44.350 align:middle line:84% So I can let N go to infinity to conclude 00:30:44.350 --> 00:30:48.610 align:middle line:84% that the outer measure of A is bigger than or equal to sum 00:30:48.610 --> 00:30:52.090 align:middle line:84% from n equals 1 to infinity of the outer measure 00:30:52.090 --> 00:31:02.650 align:middle line:84% of A intersect E n plus the outer measure of the A 00:31:02.650 --> 00:31:04.960 align:middle line:84% intersect the complement of the union. 00:31:04.960 --> 00:31:08.970 align:middle line:84% Now, remember what we proved about outer measure-- 00:31:08.970 --> 00:31:11.700 align:middle line:84% that the sum of the outer measures 00:31:11.700 --> 00:31:16.350 align:middle line:84% is bigger than or equal to the outer measure of the union. 00:31:16.350 --> 00:31:19.860 align:middle line:84% So this thing is bigger than or equal to the outer measure 00:31:19.860 --> 00:31:26.190 align:middle line:84% of the union over all N A intersect E 00:31:26.190 --> 00:31:32.980 align:middle line:90% n plus A intersect E complement. 00:31:32.980 --> 00:31:39.090 align:middle line:84% And this is just equal to plus the outer measure 00:31:39.090 --> 00:31:42.990 align:middle line:90% of A intersect E complement. 00:31:42.990 --> 00:31:48.950 align:middle line:90% 00:31:48.950 --> 00:31:56.320 align:middle line:84% So we've shown that the collection 00:31:56.320 --> 00:31:59.170 align:middle line:84% of all Lebesgue measurable subsets of R 00:31:59.170 --> 00:32:00.340 align:middle line:90% form a sigma algebra. 00:32:00.340 --> 00:32:08.230 align:middle line:90% 00:32:08.230 --> 00:32:11.170 align:middle line:90% So let me just-- 00:32:11.170 --> 00:32:15.540 align:middle line:84% maybe I should have said this at the start, 00:32:15.540 --> 00:32:17.490 align:middle line:84% but let me just pause here for a second. 00:32:17.490 --> 00:32:20.520 align:middle line:84% And maybe you're wondering why all this sigma algebra 00:32:20.520 --> 00:32:23.050 align:middle line:90% business anyways? 00:32:23.050 --> 00:32:24.570 align:middle line:84% Why have I imposed this condition? 00:32:24.570 --> 00:32:29.475 align:middle line:84% Am I just making this definition up at as I go along? 00:32:29.475 --> 00:32:34.150 align:middle line:90% 00:32:34.150 --> 00:32:36.130 align:middle line:84% But it's kind of a condition that's 00:32:36.130 --> 00:32:39.610 align:middle line:84% forced upon us based on our expectations, 00:32:39.610 --> 00:32:42.933 align:middle line:90% in the following sense-- 00:32:42.933 --> 00:32:44.350 align:middle line:84% you remember one of the properties 00:32:44.350 --> 00:32:49.030 align:middle line:84% that we wanted of Lebesgue measure, or measure in general, 00:32:49.030 --> 00:32:54.490 align:middle line:84% was that the measure of a countable union 00:32:54.490 --> 00:33:00.020 align:middle line:84% of disjoint sets is equal to the sum of the measures. 00:33:00.020 --> 00:33:05.810 align:middle line:84% And I stated without proof-- but there is a proof 00:33:05.810 --> 00:33:11.180 align:middle line:84% in the textbook-- that you cannot have a measure defined 00:33:11.180 --> 00:33:15.410 align:middle line:84% on every subset satisfying those properties that we outlined. 00:33:15.410 --> 00:33:18.410 align:middle line:84% So you have to have some collection of subsets 00:33:18.410 --> 00:33:21.650 align:middle line:84% on which you have these properties, 00:33:21.650 --> 00:33:24.590 align:middle line:84% that the measure of an interval is the length of the interval, 00:33:24.590 --> 00:33:26.900 align:middle line:84% and that's translation invariant, 00:33:26.900 --> 00:33:28.940 align:middle line:84% and that the measure of the union 00:33:28.940 --> 00:33:30.680 align:middle line:90% is the sum of the measures. 00:33:30.680 --> 00:33:35.300 align:middle line:84% That last statement, though, hidden in there, 00:33:35.300 --> 00:33:37.010 align:middle line:84% there's kind of a subtle assumption-- 00:33:37.010 --> 00:33:39.320 align:middle line:84% that if you have a measure defined 00:33:39.320 --> 00:33:43.940 align:middle line:84% on some collection of subsets, then that collection of subsets 00:33:43.940 --> 00:33:47.420 align:middle line:84% better be closed under taking countable unions. 00:33:47.420 --> 00:33:50.300 align:middle line:84% If I'm to be able to make the statement that I want, 00:33:50.300 --> 00:33:55.070 align:middle line:84% that the measure of a countable union of disjoint sets 00:33:55.070 --> 00:34:00.380 align:middle line:84% is equal to the sum of the measures, then hidden in there 00:34:00.380 --> 00:34:04.460 align:middle line:84% is that my measure has to be defined for a countable union 00:34:04.460 --> 00:34:06.350 align:middle line:90% of measurable sets. 00:34:06.350 --> 00:34:09.500 align:middle line:84% In other words, if I have a countable collection 00:34:09.500 --> 00:34:12.650 align:middle line:84% of measurable sets, then for the statement 00:34:12.650 --> 00:34:14.810 align:middle line:84% I want to even make sense, I should 00:34:14.810 --> 00:34:17.449 align:middle line:84% have that the union of this countable collection 00:34:17.449 --> 00:34:21.230 align:middle line:84% of measurable sets is contained in the class of sets 00:34:21.230 --> 00:34:23.750 align:middle line:90% that I'm measuring. 00:34:23.750 --> 00:34:31.250 align:middle line:84% And so before we even discussed outer measure, 00:34:31.250 --> 00:34:34.429 align:middle line:84% all that, how you could maybe see 00:34:34.429 --> 00:34:39.080 align:middle line:84% coming that we would have some condition like the class 00:34:39.080 --> 00:34:41.239 align:middle line:84% of sets that we're going to measure will be a sigma 00:34:41.239 --> 00:34:43.820 align:middle line:84% algebra, or should be a sigma algebra, 00:34:43.820 --> 00:34:48.389 align:middle line:84% should be closed under taking countable unions. 00:34:48.389 --> 00:34:50.897 align:middle line:84% So maybe I was rambling, but hopefully, you 00:34:50.897 --> 00:34:51.980 align:middle line:90% got something out of that. 00:34:51.980 --> 00:35:03.110 align:middle line:90% 00:35:03.110 --> 00:35:18.120 align:middle line:84% So we've shown that the set of measurable sets 00:35:18.120 --> 00:35:19.660 align:middle line:90% is a sigma algebra. 00:35:19.660 --> 00:35:22.920 align:middle line:84% Now what I'm going to show is that it contains the Borel 00:35:22.920 --> 00:35:23.830 align:middle line:90% sigma algebra. 00:35:23.830 --> 00:35:26.700 align:middle line:84% So remember, the Borel sigma algebra 00:35:26.700 --> 00:35:34.230 align:middle line:84% is the smallest sigma algebra that contains all open sets. 00:35:34.230 --> 00:35:39.020 align:middle line:84% If I have any other sigma algebra containing open sets, 00:35:39.020 --> 00:35:40.610 align:middle line:84% that contains all open sets, then 00:35:40.610 --> 00:35:44.030 align:middle line:84% it must contain the Borel sigma algebra because the Borel sigma 00:35:44.030 --> 00:35:45.070 align:middle line:90% algebra is the smallest. 00:35:45.070 --> 00:35:52.160 align:middle line:90% 00:35:52.160 --> 00:35:56.510 align:middle line:84% And I should quantify smallest-- smallest meaning with respect 00:35:56.510 --> 00:35:57.950 align:middle line:90% to inclusion. 00:35:57.950 --> 00:36:04.400 align:middle line:84% If there's any other sigma algebra that contains all 00:36:04.400 --> 00:36:06.740 align:middle line:84% open sets, then the Borel sigma algebra B 00:36:06.740 --> 00:36:08.210 align:middle line:84% is contained in that sigma algebra. 00:36:08.210 --> 00:36:13.050 align:middle line:90% 00:36:13.050 --> 00:36:18.915 align:middle line:84% But first, let's prove kind of a simpler case. 00:36:18.915 --> 00:36:23.737 align:middle line:90% 00:36:23.737 --> 00:36:25.570 align:middle line:84% So let me state this, and then I'll explain. 00:36:25.570 --> 00:36:34.130 align:middle line:84% For all a in R, the open interval a to infinity 00:36:34.130 --> 00:36:37.440 align:middle line:90% is measurable. 00:36:37.440 --> 00:36:40.470 align:middle line:84% So in the end, we want to be able to show-- 00:36:40.470 --> 00:36:42.330 align:middle line:84% so we already know M is a sigma algebra. 00:36:42.330 --> 00:36:45.550 align:middle line:84% If we can then show that every open set is measurable, 00:36:45.550 --> 00:36:50.940 align:middle line:84% then that means that M is a sigma algebra containing 00:36:50.940 --> 00:36:52.980 align:middle line:84% all open sets, and therefore, it must 00:36:52.980 --> 00:36:55.140 align:middle line:84% contain the Borel sigma algebra, which remember, 00:36:55.140 --> 00:36:59.290 align:middle line:84% is the smallest sigma algebra containing all open sets. 00:36:59.290 --> 00:37:03.570 align:middle line:84% So to build up to showing that every open set is measurable, 00:37:03.570 --> 00:37:07.820 align:middle line:84% let's start with a very simple type 00:37:07.820 --> 00:37:15.045 align:middle line:84% of open set, half open or half infinite open interval. 00:37:15.045 --> 00:37:17.770 align:middle line:90% 00:37:17.770 --> 00:37:26.540 align:middle line:84% So let A be a subset of R. And let's 00:37:26.540 --> 00:37:38.090 align:middle line:84% write A1 to be A intersect a to infinity, and A2 be A intersect 00:37:38.090 --> 00:37:45.065 align:middle line:84% the complement, which is just minus infinity to a closed. 00:37:45.065 --> 00:37:47.710 align:middle line:90% 00:37:47.710 --> 00:37:50.500 align:middle line:90% So what do we want to show? 00:37:50.500 --> 00:37:54.910 align:middle line:84% To show that this set is measurable, 00:37:54.910 --> 00:37:58.720 align:middle line:84% we want to show that the outer measure of A1 00:37:58.720 --> 00:38:02.920 align:middle line:84% plus the outer measure of A2 is less 00:38:02.920 --> 00:38:08.850 align:middle line:84% than or equal to the outer measure of A 00:38:08.850 --> 00:38:11.790 align:middle line:84% because A1 is A intersect a set, A2 00:38:11.790 --> 00:38:15.670 align:middle line:84% is A intersect a complement of that set. 00:38:15.670 --> 00:38:18.730 align:middle line:84% Now, if the outer measure of A is infinite, 00:38:18.730 --> 00:38:21.520 align:middle line:90% this holds regardless. 00:38:21.520 --> 00:38:29.820 align:middle line:84% So if the outer measure is infinite, done. 00:38:29.820 --> 00:38:36.400 align:middle line:84% So suppose the outer measure is finite. 00:38:36.400 --> 00:38:42.533 align:middle line:90% 00:38:42.533 --> 00:38:43.950 align:middle line:84% So what we're going to do is we're 00:38:43.950 --> 00:38:49.320 align:middle line:84% going to show that this sum on the left-hand side 00:38:49.320 --> 00:38:53.520 align:middle line:84% is less than or equal to this plus epsilon, where epsilon 00:38:53.520 --> 00:38:54.660 align:middle line:90% is arbitrary. 00:38:54.660 --> 00:38:58.930 align:middle line:84% And then we can send epsilon to 0 to get the inequality. 00:38:58.930 --> 00:39:03.580 align:middle line:84% And everything will reduce to what we've done with intervals, 00:39:03.580 --> 00:39:05.800 align:middle line:90% as you'll now see. 00:39:05.800 --> 00:39:27.000 align:middle line:84% So let I n be a collection of open intervals such that-- 00:39:27.000 --> 00:39:30.690 align:middle line:84% remember, the outer measure of A is the infimum 00:39:30.690 --> 00:39:34.110 align:middle line:84% of the sum of lengths of intervals covering 00:39:34.110 --> 00:39:41.790 align:middle line:84% A-- so such that sum over n length of I n is less than 00:39:41.790 --> 00:39:46.365 align:middle line:84% or equal to the outer measure of A plus epsilon. 00:39:46.365 --> 00:39:56.550 align:middle line:90% 00:39:56.550 --> 00:40:09.312 align:middle line:84% Define J n to be I n intersect A infinity K n to be I 00:40:09.312 --> 00:40:12.740 align:middle line:84% n intersect the complement of A infinity. 00:40:12.740 --> 00:40:18.790 align:middle line:90% 00:40:18.790 --> 00:40:23.380 align:middle line:90% Now, the union of the J n's-- 00:40:23.380 --> 00:40:39.600 align:middle line:84% well, first off, let me say then each of these sets J n and K n 00:40:39.600 --> 00:40:43.710 align:middle line:90% are intervals. 00:40:43.710 --> 00:40:46.620 align:middle line:84% They're the intersection of two intervals, 00:40:46.620 --> 00:40:53.115 align:middle line:84% one open and the other closed, or empty. 00:40:53.115 --> 00:40:59.840 align:middle line:90% 00:40:59.840 --> 00:41:04.370 align:middle line:84% Now, the union of the I n's covers A. 00:41:04.370 --> 00:41:07.070 align:middle line:84% So if I take the unions of the J n's, this 00:41:07.070 --> 00:41:10.550 align:middle line:84% will cover A intersect A infinity. 00:41:10.550 --> 00:41:14.000 align:middle line:90% 00:41:14.000 --> 00:41:22.690 align:middle line:84% Then A1 is contained in the union of the J n's. 00:41:22.690 --> 00:41:30.040 align:middle line:84% And similarly, A2 is contained in the union of the K n's. 00:41:30.040 --> 00:41:33.810 align:middle line:90% And one more thing-- 00:41:33.810 --> 00:41:39.110 align:middle line:84% now, I n is an interval, and it's 00:41:39.110 --> 00:41:47.170 align:middle line:84% simple to check since just based on this being a finite interval 00:41:47.170 --> 00:41:51.820 align:middle line:84% that if I take the length of I n, 00:41:51.820 --> 00:41:59.200 align:middle line:84% this is equal to the length of J n plus the length of K n. 00:41:59.200 --> 00:42:04.203 align:middle line:84% So I take each I n, and I split it up into two subintervals. 00:42:04.203 --> 00:42:06.370 align:middle line:84% They're not going to be open intervals, necessarily. 00:42:06.370 --> 00:42:08.080 align:middle line:84% This will be an open interval, this 00:42:08.080 --> 00:42:12.220 align:middle line:84% one not necessarily, but the sum of the lengths of these two 00:42:12.220 --> 00:42:16.928 align:middle line:84% intervals will add up to the length of the interval I n. 00:42:16.928 --> 00:42:17.470 align:middle line:90% That's clear. 00:42:17.470 --> 00:42:25.322 align:middle line:90% 00:42:25.322 --> 00:42:42.430 align:middle line:84% There's, if you'd like, J n and K n will be including that. 00:42:42.430 --> 00:42:46.040 align:middle line:90% 00:42:46.040 --> 00:42:50.590 align:middle line:90% And now, we're almost home free. 00:42:50.590 --> 00:42:52.030 align:middle line:84% So A1 is contained in this union. 00:42:52.030 --> 00:42:53.590 align:middle line:90% A2 is contained in this union. 00:42:53.590 --> 00:43:01.310 align:middle line:84% So if I look at the measure of A1 plus the measure of A2, 00:43:01.310 --> 00:43:05.660 align:middle line:84% the measure of A1, since it's contained in this union, 00:43:05.660 --> 00:43:08.150 align:middle line:84% is going to be less than or equal to the sum of the outer 00:43:08.150 --> 00:43:14.000 align:middle line:90% measures of J n's. 00:43:14.000 --> 00:43:19.350 align:middle line:84% And the measure of A2, again, is contained in K, 00:43:19.350 --> 00:43:22.950 align:middle line:84% in the union of the K n's, so that's less than or equal 00:43:22.950 --> 00:43:25.740 align:middle line:84% to the sum of the measures of the K n's. 00:43:25.740 --> 00:43:30.570 align:middle line:84% And now, bringing these two together in one sum over n, 00:43:30.570 --> 00:43:34.510 align:middle line:84% and using the fact that this is equal to-- 00:43:34.510 --> 00:43:36.280 align:middle line:84% so the outer measure of an interval 00:43:36.280 --> 00:43:40.770 align:middle line:84% is equal to its length, which we proved last time-- 00:43:40.770 --> 00:43:43.910 align:middle line:90% 00:43:43.910 --> 00:43:49.870 align:middle line:84% this is equal to sum length of the I n. 00:43:49.870 --> 00:43:51.880 align:middle line:84% And remember, how did we choose the I n? 00:43:51.880 --> 00:43:55.990 align:middle line:84% We chose this I n so that some of the lengths of the I n 00:43:55.990 --> 00:43:58.300 align:middle line:84% is less than or equal to the outer measure 00:43:58.300 --> 00:43:59.710 align:middle line:90% of A plus epsilon. 00:43:59.710 --> 00:44:06.740 align:middle line:90% 00:44:06.740 --> 00:44:12.620 align:middle line:84% So I should have added here, let epsilon be positive. 00:44:12.620 --> 00:44:13.760 align:middle line:90% Sorry about that. 00:44:13.760 --> 00:44:18.190 align:middle line:90% 00:44:18.190 --> 00:44:26.210 align:middle line:84% So I've shown that for arbitrary epsilon positive, 00:44:26.210 --> 00:44:29.598 align:middle line:84% this number over here is less than or equal to-- 00:44:29.598 --> 00:44:30.640 align:middle line:90% I still haven't finished. 00:44:30.640 --> 00:44:33.037 align:middle line:90% Sorry, getting ahead of myself. 00:44:33.037 --> 00:44:34.745 align:middle line:84% That's what happens when you get excited. 00:44:34.745 --> 00:44:37.525 align:middle line:90% 00:44:37.525 --> 00:44:39.650 align:middle line:84% And this is less than or equal to the outer measure 00:44:39.650 --> 00:44:41.615 align:middle line:90% of A plus epsilon. 00:44:41.615 --> 00:44:44.690 align:middle line:90% 00:44:44.690 --> 00:44:48.300 align:middle line:84% So we've shown that for arbitrary epsilon positive, 00:44:48.300 --> 00:44:50.990 align:middle line:84% this number is less than or equal to this number 00:44:50.990 --> 00:44:51.930 align:middle line:90% plus epsilon. 00:44:51.930 --> 00:44:54.710 align:middle line:84% So I can send epsilon to 0 to get 00:44:54.710 --> 00:45:02.120 align:middle line:84% that the outer measure of A1 plus the outer measure of A2 00:45:02.120 --> 00:45:05.058 align:middle line:84% is less than or equal to the outer measure of A, which 00:45:05.058 --> 00:45:06.725 align:middle line:84% remember, that's what we wanted to show. 00:45:06.725 --> 00:45:22.050 align:middle line:90% 00:45:22.050 --> 00:45:27.630 align:middle line:84% So we've shown that these open intervals from A to infinity 00:45:27.630 --> 00:45:30.000 align:middle line:90% are measurable. 00:45:30.000 --> 00:45:44.600 align:middle line:84% It's now not a long trip to saying that every open set is 00:45:44.600 --> 00:45:45.740 align:middle line:90% Lebesgue measurable. 00:45:45.740 --> 00:45:48.410 align:middle line:90% 00:45:48.410 --> 00:46:07.680 align:middle line:84% So theorem-- every open set is Lebesgue measurable, 00:46:07.680 --> 00:46:12.120 align:middle line:84% and thus, a Borel sigma algebra, which is the smallest sigma 00:46:12.120 --> 00:46:14.280 align:middle line:84% algebra containing all open sets, 00:46:14.280 --> 00:46:17.085 align:middle line:84% is contained in the sigma algebra of measurable sets. 00:46:17.085 --> 00:46:32.190 align:middle line:90% 00:46:32.190 --> 00:46:35.340 align:middle line:84% So we've shown these types of intervals are open. 00:46:35.340 --> 00:46:38.790 align:middle line:84% Let's show finite open intervals are also-- 00:46:38.790 --> 00:46:40.890 align:middle line:84% I mean, we've shown these types of open intervals 00:46:40.890 --> 00:46:41.850 align:middle line:90% are measurable. 00:46:41.850 --> 00:46:49.800 align:middle line:84% Let's now show that finite open intervals are measurable. 00:46:49.800 --> 00:47:06.630 align:middle line:84% So now, we have for all b in R that minus infinity to b, this 00:47:06.630 --> 00:47:13.488 align:middle line:84% is equal to the union n equals 1 to infinity of what? 00:47:13.488 --> 00:47:22.700 align:middle line:84% Of the union of these half closed 00:47:22.700 --> 00:47:33.970 align:middle line:84% intervals, which we can write as the complement of these types 00:47:33.970 --> 00:47:36.400 align:middle line:90% of open intervals. 00:47:36.400 --> 00:47:42.890 align:middle line:84% Now, we just showed that these types of intervals are open, 00:47:42.890 --> 00:47:45.650 align:middle line:84% I mean, these types of open intervals are measurable. 00:47:45.650 --> 00:47:47.930 align:middle line:84% Therefore, their complement is measurable 00:47:47.930 --> 00:47:52.380 align:middle line:84% and the collection of measurable sets is a sigma algebra. 00:47:52.380 --> 00:47:54.800 align:middle line:84% So countable unions are also measurable. 00:47:54.800 --> 00:48:01.420 align:middle line:84% So we conclude that this guy is measurable. 00:48:01.420 --> 00:48:14.130 align:middle line:90% 00:48:14.130 --> 00:48:21.875 align:middle line:84% And therefore, remember for an algebra, it's closed under-- 00:48:21.875 --> 00:48:24.250 align:middle line:84% for a sigma algebra, it's closed under taking complements 00:48:24.250 --> 00:48:26.080 align:middle line:90% and taking countable unions. 00:48:26.080 --> 00:48:29.860 align:middle line:84% But by De Morgan's laws, that means 00:48:29.860 --> 00:48:33.880 align:middle line:84% it's also closed under taking intersections. 00:48:33.880 --> 00:48:37.750 align:middle line:84% We conclude that for all a, b, and R, 00:48:37.750 --> 00:48:41.590 align:middle line:84% any finite open interval a, b, which 00:48:41.590 --> 00:48:50.930 align:middle line:84% is equal to minus infinity to b intersect a to infinity, 00:48:50.930 --> 00:48:53.140 align:middle line:84% this is measurable by what we just proved. 00:48:53.140 --> 00:48:55.810 align:middle line:84% This is measurable by the theorem we proved before. 00:48:55.810 --> 00:48:58.450 align:middle line:84% And sigma algebras are also closed 00:48:58.450 --> 00:48:59.890 align:middle line:90% under taking countable unions. 00:48:59.890 --> 00:49:01.330 align:middle line:90% This is just a finite union. 00:49:01.330 --> 00:49:04.660 align:middle line:90% So this is also measurable. 00:49:04.660 --> 00:49:15.310 align:middle line:90% 00:49:15.310 --> 00:49:24.795 align:middle line:84% Now, maybe you covered this in 100B, maybe you didn't. 00:49:24.795 --> 00:49:26.420 align:middle line:84% It's going to appear on the assignment. 00:49:26.420 --> 00:49:30.560 align:middle line:84% But I'll underline what's going to be on the assignment. 00:49:30.560 --> 00:49:48.490 align:middle line:84% But since every open subset of R is a countable union of, 00:49:48.490 --> 00:49:57.710 align:middle line:84% in fact, disjoint open intervals-- 00:49:57.710 --> 00:50:00.070 align:middle line:84% open intervals meaning it could be a finite one, 00:50:00.070 --> 00:50:01.820 align:middle line:84% it could be one like that, it could be one 00:50:01.820 --> 00:50:04.820 align:middle line:84% like we wrote over there where it's a to infinity-- 00:50:04.820 --> 00:50:09.290 align:middle line:84% but every open subset can be written as a countable union 00:50:09.290 --> 00:50:11.120 align:middle line:90% of disjoint open intervals. 00:50:11.120 --> 00:50:14.870 align:middle line:84% And since open intervals, we've now concluded, 00:50:14.870 --> 00:50:18.020 align:middle line:84% are always measurable, that means 00:50:18.020 --> 00:50:19.490 align:middle line:90% their union is measurable. 00:50:19.490 --> 00:50:30.220 align:middle line:90% 00:50:30.220 --> 00:50:32.500 align:middle line:84% We conclude that every open set is measurable. 00:50:32.500 --> 00:50:50.500 align:middle line:90% 00:50:50.500 --> 00:50:57.130 align:middle line:84% We now have a collection that we've said are measurable sets. 00:50:57.130 --> 00:50:58.450 align:middle line:90% They form a sigma algebra. 00:50:58.450 --> 00:51:02.060 align:middle line:84% They contain the Borel sigma algebra, 00:51:02.060 --> 00:51:05.170 align:middle line:90% which contains all open sets. 00:51:05.170 --> 00:51:12.600 align:middle line:90% 00:51:12.600 --> 00:51:16.920 align:middle line:84% So now, we define Lebesgue measure. 00:51:16.920 --> 00:51:37.180 align:middle line:84% If E is a measurable set, the Lebesgue measure of E 00:51:37.180 --> 00:51:43.810 align:middle line:84% is denoted by m of E. And it's just given 00:51:43.810 --> 00:51:49.710 align:middle line:90% by the outer measure of E. 00:51:49.710 --> 00:51:53.700 align:middle line:84% So like I said when we first started all this business, 00:51:53.700 --> 00:51:57.060 align:middle line:84% Lebesgue measure is simply outer measure 00:51:57.060 --> 00:52:03.630 align:middle line:84% restricted to a collection of well-behaved sets. 00:52:03.630 --> 00:52:06.480 align:middle line:84% And you see that here, is that Lebesgue measure is nothing 00:52:06.480 --> 00:52:10.620 align:middle line:84% but outer measure restricted to those sets which 00:52:10.620 --> 00:52:11.610 align:middle line:90% we call measurable. 00:52:11.610 --> 00:52:14.100 align:middle line:84% And measurable meant that they had this property 00:52:14.100 --> 00:52:18.207 align:middle line:84% that they kind of split sets evenly with respect 00:52:18.207 --> 00:52:18.915 align:middle line:90% to outer measure. 00:52:18.915 --> 00:52:25.780 align:middle line:90% 00:52:25.780 --> 00:52:36.000 align:middle line:84% So immediately, we get to a few simple things. 00:52:36.000 --> 00:52:43.930 align:middle line:84% Theorem-- if A and B are measurable, 00:52:43.930 --> 00:52:50.050 align:middle line:84% and A is contained in B, then the measure of A 00:52:50.050 --> 00:52:53.140 align:middle line:84% is less than or equal to the measure of B. Why is this? 00:52:53.140 --> 00:52:56.530 align:middle line:84% Again, because outer measure satisfies that. 00:52:56.530 --> 00:52:58.420 align:middle line:84% m of A is just the outer measure of A, 00:52:58.420 --> 00:53:01.060 align:middle line:84% and that's less than or equal to the outer measure of B, which 00:53:01.060 --> 00:53:02.890 align:middle line:84% is, by definition, the measure of B. 00:53:02.890 --> 00:53:06.400 align:middle line:84% So the point is that Lebesgue measure 00:53:06.400 --> 00:53:08.680 align:middle line:84% inherits many properties from outer measure. 00:53:08.680 --> 00:53:10.520 align:middle line:90% In particular, we have this. 00:53:10.520 --> 00:53:13.360 align:middle line:84% And then we also have, since we know-- 00:53:13.360 --> 00:53:22.820 align:middle line:90% 00:53:22.820 --> 00:53:26.390 align:middle line:90% let me make one more. 00:53:26.390 --> 00:53:37.240 align:middle line:90% 00:53:37.240 --> 00:53:43.370 align:middle line:90% Every interval is measurable. 00:53:43.370 --> 00:53:44.990 align:middle line:90% Let's state it this way. 00:53:44.990 --> 00:53:49.330 align:middle line:90% 00:53:49.330 --> 00:54:05.570 align:middle line:84% If I is an interval, then I is measurable and the measure of I 00:54:05.570 --> 00:54:13.580 align:middle line:84% equals the length of I. So we've shown all open intervals are 00:54:13.580 --> 00:54:14.130 align:middle line:90% measurable. 00:54:14.130 --> 00:54:16.130 align:middle line:84% That's what we did a minute ago when we showed-- 00:54:16.130 --> 00:54:20.300 align:middle line:90% 00:54:20.300 --> 00:54:23.300 align:middle line:84% so first off, let me not get ahead of myself. 00:54:23.300 --> 00:54:26.540 align:middle line:84% If I've shown that every I is measurable, then 00:54:26.540 --> 00:54:30.380 align:middle line:84% since Lebesgue measure is just a restriction of outer measure, 00:54:30.380 --> 00:54:32.510 align:middle line:84% and the outer measure of an interval 00:54:32.510 --> 00:54:35.450 align:middle line:84% is the length of the interval, this is immediate. 00:54:35.450 --> 00:54:42.920 align:middle line:84% So I just need to verify that every interval is measurable. 00:54:42.920 --> 00:54:46.910 align:middle line:84% Now, we've shown every open interval is measurable, 00:54:46.910 --> 00:54:53.360 align:middle line:84% but from there, it's not difficult to get 00:54:53.360 --> 00:54:56.070 align:middle line:84% that every interval is measurable. 00:54:56.070 --> 00:55:02.460 align:middle line:84% So for example, if I take a closed and bounded interval, 00:55:02.460 --> 00:55:04.810 align:middle line:90% this is equal to-- 00:55:04.810 --> 00:55:07.700 align:middle line:90% what is this equal to? 00:55:07.700 --> 00:55:14.950 align:middle line:84% This is equal to b infinity complement intersect minus 00:55:14.950 --> 00:55:17.230 align:middle line:90% infinity a complement. 00:55:17.230 --> 00:55:20.140 align:middle line:90% 00:55:20.140 --> 00:55:26.860 align:middle line:84% So b infinity complement gives me b to infinity, including b. 00:55:26.860 --> 00:55:29.170 align:middle line:84% The complement of minus infinity to a 00:55:29.170 --> 00:55:32.800 align:middle line:84% is a to infinity, including a, and their intersection 00:55:32.800 --> 00:55:33.760 align:middle line:90% gives me a, b. 00:55:33.760 --> 00:55:36.520 align:middle line:84% Now, open intervals are measurable. 00:55:36.520 --> 00:55:38.540 align:middle line:84% Therefore, the complement is measurable. 00:55:38.540 --> 00:55:40.630 align:middle line:84% So each of these things is measurable, 00:55:40.630 --> 00:55:42.980 align:middle line:84% and therefore, their intersection is measurable. 00:55:42.980 --> 00:55:47.260 align:middle line:90% So that's measurable. 00:55:47.260 --> 00:55:53.920 align:middle line:84% And it's the same game if I take away one of these guys, 00:55:53.920 --> 00:55:59.250 align:middle line:84% except now, we would use kind of a trick like that. 00:55:59.250 --> 00:56:02.910 align:middle line:90% So let me do one of them. 00:56:02.910 --> 00:56:08.230 align:middle line:90% Let's say we look at a, b. 00:56:08.230 --> 00:56:10.982 align:middle line:90% 00:56:10.982 --> 00:56:12.440 align:middle line:84% Well, I mean, I don't think we even 00:56:12.440 --> 00:56:13.690 align:middle line:90% have to do anything like that. 00:56:13.690 --> 00:56:17.684 align:middle line:84% Let's say I look at something like this-- 00:56:17.684 --> 00:56:31.620 align:middle line:84% this is equal to b intersect complement. 00:56:31.620 --> 00:56:35.520 align:middle line:84% So a half-open interval including a, not including b, 00:56:35.520 --> 00:56:39.555 align:middle line:84% is equal to minus infinity b intersect minus infinity 00:56:39.555 --> 00:56:40.890 align:middle line:90% a complement. 00:56:40.890 --> 00:56:45.255 align:middle line:84% Because then, this gives me a to infinity, including a. 00:56:45.255 --> 00:56:47.757 align:middle line:84% And this is measurable, this is measurable, 00:56:47.757 --> 00:56:50.340 align:middle line:84% the complement's measurable, and therefore, the intersection's 00:56:50.340 --> 00:56:50.840 align:middle line:90% measurable. 00:56:50.840 --> 00:56:52.980 align:middle line:90% So this is measurable. 00:56:52.980 --> 00:56:58.050 align:middle line:84% And those are the only two examples I'm going to do. 00:56:58.050 --> 00:57:01.020 align:middle line:84% And basically, by taking the complements of these, 00:57:01.020 --> 00:57:04.620 align:middle line:84% you get the other types of intervals. 00:57:04.620 --> 00:57:09.520 align:middle line:84% So this is a good one because this is one of the properties, 00:57:09.520 --> 00:57:14.860 align:middle line:84% remember, that we wanted of measure. 00:57:14.860 --> 00:57:17.380 align:middle line:84% We at least wanted to be able to measure intervals. 00:57:17.380 --> 00:57:20.290 align:middle line:84% And we wanted the measure of an interval, 00:57:20.290 --> 00:57:22.390 align:middle line:84% or at least a Lebesgue measure of an interval, 00:57:22.390 --> 00:57:24.715 align:middle line:84% to be the length of that interval. 00:57:24.715 --> 00:57:28.070 align:middle line:90% 00:57:28.070 --> 00:57:38.340 align:middle line:84% And now, let's verify that other condition that we wanted, 00:57:38.340 --> 00:57:45.190 align:middle line:84% that the measure of a countable disjoint union 00:57:45.190 --> 00:57:47.300 align:middle line:90% is the sum of the measures. 00:57:47.300 --> 00:58:11.170 align:middle line:84% So suppose E n is a countable collection 00:58:11.170 --> 00:58:16.085 align:middle line:90% of disjoint measurable sets. 00:58:16.085 --> 00:58:20.180 align:middle line:90% 00:58:20.180 --> 00:58:25.460 align:middle line:84% Then the Lebesgue measure of the union 00:58:25.460 --> 00:58:36.130 align:middle line:84% is equal to the sum of the Lebesgue measure of the sets. 00:58:36.130 --> 00:58:41.350 align:middle line:84% Now this, we have one of the inequalities here 00:58:41.350 --> 00:58:45.100 align:middle line:84% or we always have this is less than or equal to this 00:58:45.100 --> 00:58:47.980 align:middle line:84% simply because that follows from outer measure. 00:58:47.980 --> 00:58:51.130 align:middle line:84% Remember, outer measure satisfied this 00:58:51.130 --> 00:58:55.180 align:middle line:84% with an inequality there, with M star and M star, but not 00:58:55.180 --> 00:58:56.780 align:middle line:90% equality. 00:58:56.780 --> 00:58:59.210 align:middle line:84% Specializing to these well-behaved sets 00:58:59.210 --> 00:59:01.430 align:middle line:90% gives us equality, as we'll see. 00:59:01.430 --> 00:59:08.300 align:middle line:90% 00:59:08.300 --> 00:59:14.900 align:middle line:84% So let me just reiterate that we do have one-- 00:59:14.900 --> 00:59:17.000 align:middle line:84% we'll prove this by showing one is less than 00:59:17.000 --> 00:59:18.950 align:middle line:84% or equal to the other and vice versa. 00:59:18.950 --> 00:59:22.700 align:middle line:84% We always have this is less than or equal to this 00:59:22.700 --> 00:59:23.900 align:middle line:90% because of outer measure. 00:59:23.900 --> 00:59:27.920 align:middle line:90% 00:59:27.920 --> 00:59:37.370 align:middle line:84% So first off, we know that we have 00:59:37.370 --> 00:59:39.410 align:middle line:84% this countable union is measurable 00:59:39.410 --> 00:59:42.140 align:middle line:90% because M is a sigma algebra. 00:59:42.140 --> 00:59:43.760 align:middle line:90% We showed that already. 00:59:43.760 --> 00:59:50.630 align:middle line:84% And therefore, the measure of the union is, by definition, 00:59:50.630 --> 00:59:55.050 align:middle line:84% equal to the outer measure of the union, 00:59:55.050 --> 01:00:01.800 align:middle line:84% which is less than or equal to the sum of the outer measures 01:00:01.800 --> 01:00:02.970 align:middle line:90% of the E n's. 01:00:02.970 --> 01:00:05.040 align:middle line:84% And remember, these are all measurable. 01:00:05.040 --> 01:00:08.385 align:middle line:84% So by definition, the outer measure is the measure. 01:00:08.385 --> 01:00:11.840 align:middle line:90% 01:00:11.840 --> 01:00:14.560 align:middle line:84% So this is what I meant by we always have one inequality 01:00:14.560 --> 01:00:18.400 align:middle line:84% or we already have one side of this inequality 01:00:18.400 --> 01:00:20.840 align:middle line:90% from outer measure. 01:00:20.840 --> 01:00:25.960 align:middle line:84% So now we just need to show the opposite inequality. 01:00:25.960 --> 01:00:31.560 align:middle line:90% 01:00:31.560 --> 01:00:37.755 align:middle line:84% We now show sum n measure of E n is less than or equal to-- 01:00:37.755 --> 01:00:46.930 align:middle line:90% 01:00:46.930 --> 01:00:49.555 align:middle line:90% so how do we show this? 01:00:49.555 --> 01:00:55.090 align:middle line:90% 01:00:55.090 --> 01:00:56.330 align:middle line:90% Let m be a natural number. 01:00:56.330 --> 01:01:00.630 align:middle line:90% 01:01:00.630 --> 01:01:23.900 align:middle line:84% Then what is the measure of a finite union of these guys? 01:01:23.900 --> 01:01:28.080 align:middle line:84% Well, I can write this also as the measure. 01:01:28.080 --> 01:01:32.180 align:middle line:84% So this is measurable, and therefore, I 01:01:32.180 --> 01:01:36.140 align:middle line:84% can write this as, in fact, R intersect. 01:01:36.140 --> 01:01:39.750 align:middle line:84% This is kind of a stupid way to write it, 01:01:39.750 --> 01:01:41.270 align:middle line:84% but I'm just doing it this way so 01:01:41.270 --> 01:01:45.230 align:middle line:84% that it looks like something we've already proven. 01:01:45.230 --> 01:01:49.350 align:middle line:84% Earlier, we proved that for measurable sets, 01:01:49.350 --> 01:01:53.730 align:middle line:84% disjoint measurable sets, the outer measure of a set 01:01:53.730 --> 01:01:57.990 align:middle line:84% intersect a finite union of disjoint measurable sets 01:01:57.990 --> 01:02:04.260 align:middle line:84% is equal to the sum of these measures. 01:02:04.260 --> 01:02:10.160 align:middle line:90% 01:02:10.160 --> 01:02:13.610 align:middle line:90% And this is equal to E sub n. 01:02:13.610 --> 01:02:15.590 align:middle line:84% And the outer measure of a measurable set 01:02:15.590 --> 01:02:18.050 align:middle line:84% is, by definition again, the measure of the set. 01:02:18.050 --> 01:02:25.590 align:middle line:84% So this is equal to n equals 1 to E sub n. 01:02:25.590 --> 01:02:29.040 align:middle line:84% So what we've shown is that in fact, for a finite disjoint 01:02:29.040 --> 01:02:32.970 align:middle line:84% union, the measure is equal to the sum of the measures. 01:02:32.970 --> 01:02:35.595 align:middle line:84% I mean, we already did prove that, except with an A there. 01:02:35.595 --> 01:02:38.340 align:middle line:90% 01:02:38.340 --> 01:02:41.090 align:middle line:84% But this gives us the opposite inequality 01:02:41.090 --> 01:02:52.000 align:middle line:84% once we realize that we have this sum which 01:02:52.000 --> 01:03:01.380 align:middle line:84% is equal to the measure of, and which 01:03:01.380 --> 01:03:04.965 align:middle line:84% is less than or equal to the measure of the big total union. 01:03:04.965 --> 01:03:08.320 align:middle line:90% 01:03:08.320 --> 01:03:11.950 align:middle line:84% Because this set is contained in this set. 01:03:11.950 --> 01:03:17.310 align:middle line:84% So what I have is that N was arbitrary. 01:03:17.310 --> 01:03:21.030 align:middle line:84% I had that thing on the left-hand side is less than 01:03:21.030 --> 01:03:26.770 align:middle line:84% or equal to this thing on the right-hand side for arbitrary 01:03:26.770 --> 01:03:33.730 align:middle line:84% N. So now I let N go to infinity to conclude that-- 01:03:33.730 --> 01:03:42.750 align:middle line:90% 01:03:42.750 --> 01:03:43.455 align:middle line:90% as desired. 01:03:43.455 --> 01:03:47.710 align:middle line:90% 01:03:47.710 --> 01:03:50.290 align:middle line:84% So this is that other property that we wanted, 01:03:50.290 --> 01:03:56.150 align:middle line:84% that the measure of a disjoint union 01:03:56.150 --> 01:03:58.550 align:middle line:90% is the sum of the measures. 01:03:58.550 --> 01:04:00.050 align:middle line:84% Now, there was that last property 01:04:00.050 --> 01:04:06.410 align:middle line:84% we wanted that measure is translation invariant. 01:04:06.410 --> 01:04:10.595 align:middle line:84% That will be in the assignment, so let me state it here. 01:04:10.595 --> 01:04:13.900 align:middle line:90% 01:04:13.900 --> 01:04:17.760 align:middle line:84% So what you'll prove in the assignment is 01:04:17.760 --> 01:04:24.780 align:middle line:84% if E is a measurable set and x is in R, then 01:04:24.780 --> 01:04:31.020 align:middle line:84% the shift of the set x E plus x, which is the set of all y 01:04:31.020 --> 01:04:38.790 align:middle line:84% plus x such that y is in E, is measurable. 01:04:38.790 --> 01:04:47.110 align:middle line:84% And measure of E equals measure of E plus x. 01:04:47.110 --> 01:04:57.770 align:middle line:84% So this will be in the next assignment, which 01:04:57.770 --> 01:05:04.300 align:middle line:84% will be assignment, I think, 4, which is the third property 01:05:04.300 --> 01:05:05.990 align:middle line:90% that we desired. 01:05:05.990 --> 01:05:11.230 align:middle line:84% So Lebesgue measure, which is just outer measure restricted 01:05:11.230 --> 01:05:14.920 align:middle line:84% to the class of measurable subsets, which is a sigma 01:05:14.920 --> 01:05:21.180 align:middle line:84% algebra, satisfies the three major things 01:05:21.180 --> 01:05:23.340 align:middle line:90% we wanted of a measure. 01:05:23.340 --> 01:05:26.970 align:middle line:84% Unfortunately, the measure is not defined on all subsets, 01:05:26.970 --> 01:05:33.220 align:middle line:84% but it's defined on a large class of-- 01:05:33.220 --> 01:05:36.220 align:middle line:84% a very rich class of subsets of real numbers 01:05:36.220 --> 01:05:41.980 align:middle line:84% because it contains open sets, closed sets, and like I said, 01:05:41.980 --> 01:05:46.263 align:middle line:84% sigma algebras are closed under taking countable intersections 01:05:46.263 --> 01:05:46.930 align:middle line:90% and complements. 01:05:46.930 --> 01:05:48.910 align:middle line:84% So you could take a collection of open sets 01:05:48.910 --> 01:05:53.800 align:middle line:84% and take its intersection, which is not necessarily an open set, 01:05:53.800 --> 01:05:55.730 align:middle line:84% but that would be in the sigma algebra. 01:05:55.730 --> 01:05:58.150 align:middle line:84% And then you could take a countable union 01:05:58.150 --> 01:06:01.852 align:middle line:84% of those types of sets and still remain in the sigma algebra. 01:06:01.852 --> 01:06:03.310 align:middle line:84% And then you could take complements 01:06:03.310 --> 01:06:06.830 align:middle line:84% of those types of sets and stay in the sigma algebra. 01:06:06.830 --> 01:06:10.090 align:middle line:84% So like my instructor said, if you can write down the set, 01:06:10.090 --> 01:06:11.590 align:middle line:90% chances are it's measurable. 01:06:11.590 --> 01:06:16.140 align:middle line:90% 01:06:16.140 --> 01:06:23.190 align:middle line:84% So one last theorem we'll prove about measure, 01:06:23.190 --> 01:06:28.030 align:middle line:84% and then we'll call it a day, and call it 01:06:28.030 --> 01:06:31.970 align:middle line:84% for the theory of measure by itself. 01:06:31.970 --> 01:06:36.370 align:middle line:84% Then we'll move on to measurable functions, 01:06:36.370 --> 01:06:40.840 align:middle line:84% and then Lebesgue integration of measurable functions 01:06:40.840 --> 01:06:52.800 align:middle line:84% is the following, if you like continuity of measure, 01:06:52.800 --> 01:06:54.120 align:middle line:90% which is the following. 01:06:54.120 --> 01:07:12.990 align:middle line:84% So suppose E k is a collection of measurable sets 01:07:12.990 --> 01:07:20.540 align:middle line:84% such that E1 is contained in E2 is contained in E3, and so on. 01:07:20.540 --> 01:07:23.990 align:middle line:90% 01:07:23.990 --> 01:07:37.250 align:middle line:84% Then the measure of k from 1 to infinity of E k-- 01:07:37.250 --> 01:07:45.650 align:middle line:84% this is equal to limit as n goes to infinity of the measure of k 01:07:45.650 --> 01:07:55.080 align:middle line:84% equals n of E k, which equals limit as n goes to infinity 01:07:55.080 --> 01:07:58.200 align:middle line:90% of the measure of E n. 01:07:58.200 --> 01:08:01.460 align:middle line:90% 01:08:01.460 --> 01:08:08.460 align:middle line:84% Now, I just need to show this is equal to this. 01:08:08.460 --> 01:08:12.720 align:middle line:84% And that's what I'll show, is that this is equal to this. 01:08:12.720 --> 01:08:14.580 align:middle line:84% The fact that this is equal to this 01:08:14.580 --> 01:08:18.960 align:middle line:84% follows from the assumption that they're nested. 01:08:18.960 --> 01:08:21.779 align:middle line:84% E1 is contained in E2 is contained in up to E n, 01:08:21.779 --> 01:08:25.080 align:middle line:90% so the union is equal to E n. 01:08:25.080 --> 01:08:30.040 align:middle line:84% So I'm just going to show that these two things underlined 01:08:30.040 --> 01:08:31.990 align:middle line:84% in yellow are equal to each other. 01:08:31.990 --> 01:08:34.720 align:middle line:84% And then the fact that this is equal to that just follows 01:08:34.720 --> 01:08:36.250 align:middle line:90% from this assumption here. 01:08:36.250 --> 01:08:42.420 align:middle line:90% 01:08:42.420 --> 01:08:46.770 align:middle line:84% So for the proof, I'm going to do this trick again 01:08:46.770 --> 01:08:52.080 align:middle line:84% where we're going to write the union, this countable union, 01:08:52.080 --> 01:08:59.229 align:middle line:90% as a union of disjoint sets. 01:08:59.229 --> 01:09:10.450 align:middle line:84% So we let F1 equal to be E1 F k plus 1 equal to E k plus 1 01:09:10.450 --> 01:09:15.354 align:middle line:84% take away E k for k bigger than or equal to 1. 01:09:15.354 --> 01:09:23.270 align:middle line:90% 01:09:23.270 --> 01:09:29.270 align:middle line:84% And let me just remark, this is equal to E k plus 1 intersect E 01:09:29.270 --> 01:09:31.970 align:middle line:84% k complement, since they're nested, 01:09:31.970 --> 01:09:36.319 align:middle line:84% since E1 is contained in E2 is contained, and so on. 01:09:36.319 --> 01:09:40.279 align:middle line:84% And note that since the E k's are measurable, 01:09:40.279 --> 01:09:42.162 align:middle line:84% this is measurable, this is measurable. 01:09:42.162 --> 01:09:43.370 align:middle line:90% Its complement is measurable. 01:09:43.370 --> 01:09:44.790 align:middle line:90% The intersection is measurable. 01:09:44.790 --> 01:09:46.939 align:middle line:90% So each of these are measurable. 01:09:46.939 --> 01:09:50.010 align:middle line:90% 01:09:50.010 --> 01:10:06.185 align:middle line:84% Then, F k is a disjoint collection of measurable sets. 01:10:06.185 --> 01:10:11.500 align:middle line:90% 01:10:11.500 --> 01:10:18.610 align:middle line:84% And for all n in N, if I look at the union k equals 1 to n 01:10:18.610 --> 01:10:21.160 align:middle line:90% of F k-- 01:10:21.160 --> 01:10:24.100 align:middle line:90% so how am I building these guys? 01:10:24.100 --> 01:10:25.910 align:middle line:90% I take F1 to be E1. 01:10:25.910 --> 01:10:28.450 align:middle line:84% F2 is going to be E2 take away whatever 01:10:28.450 --> 01:10:33.910 align:middle line:90% was already contained in E1. 01:10:33.910 --> 01:10:37.270 align:middle line:84% F3 is whatever E3 is take away everything that was already 01:10:37.270 --> 01:10:38.470 align:middle line:90% appearing in E2. 01:10:38.470 --> 01:10:44.650 align:middle line:84% So you can check that the union from k equals 1 to n of F k 01:10:44.650 --> 01:11:03.140 align:middle line:84% is equal to E n, and therefore also, that the union is 01:11:03.140 --> 01:11:07.855 align:middle line:90% equal to this union. 01:11:07.855 --> 01:11:26.820 align:middle line:90% 01:11:26.820 --> 01:11:31.200 align:middle line:84% And now, we conclude that if I take the measure of k 01:11:31.200 --> 01:11:42.460 align:middle line:84% equals 1 to infinity of E k, this is equal to this union 01:11:42.460 --> 01:11:45.830 align:middle line:84% here, which is a union of disjoint measurable sets. 01:11:45.830 --> 01:11:48.490 align:middle line:84% So this is equal to the sum of k equals 1 01:11:48.490 --> 01:11:54.070 align:middle line:84% to infinity of the measure of F sub k, which 01:11:54.070 --> 01:11:58.840 align:middle line:84% is equal to the limit as n goes to infinity 01:11:58.840 --> 01:12:03.100 align:middle line:90% k equals 1 to n measure of F k. 01:12:03.100 --> 01:12:09.090 align:middle line:84% And again, if you like, I can rewrite this 01:12:09.090 --> 01:12:15.780 align:middle line:84% as limit as n goes to infinity, this finite sum of measures 01:12:15.780 --> 01:12:21.690 align:middle line:84% of these disjoint sets, as the measure of k equals 1 to n 01:12:21.690 --> 01:12:31.380 align:middle line:84% of F k which equals, as we noted right here, 01:12:31.380 --> 01:12:35.710 align:middle line:90% is equal to the measure of E n. 01:12:35.710 --> 01:12:44.100 align:middle line:90% 01:12:44.100 --> 01:12:49.020 align:middle line:84% So that takes care of the definition of Lebesgue measure. 01:12:49.020 --> 01:12:57.450 align:middle line:84% Next time, we will define Lebesgue measurable functions, 01:12:57.450 --> 01:13:06.990 align:middle line:90% which are, in a certain sense-- 01:13:06.990 --> 01:13:09.480 align:middle line:84% with respect to integration-- the analog 01:13:09.480 --> 01:13:11.110 align:middle line:90% of continuous functions. 01:13:11.110 --> 01:13:14.550 align:middle line:84% So continuous functions have the property 01:13:14.550 --> 01:13:22.510 align:middle line:84% that if I have an open set in the target, 01:13:22.510 --> 01:13:25.560 align:middle line:84% so if F is a function going from x to y, 01:13:25.560 --> 01:13:28.050 align:middle line:84% if I have a open set in y, then the inverse image 01:13:28.050 --> 01:13:31.230 align:middle line:84% of that open set is an open set in x. 01:13:31.230 --> 01:13:35.320 align:middle line:84% Measurable functions will be similar to that, 01:13:35.320 --> 01:13:39.360 align:middle line:84% except now with measurable sets, but not quite. 01:13:39.360 --> 01:13:42.630 align:middle line:84% We're not going to ask that they take Lebesgue measurable sets 01:13:42.630 --> 01:13:46.350 align:middle line:84% to Lebesgue measurable sets, but Borel measurable sets 01:13:46.350 --> 01:13:53.050 align:middle line:84% to the inverse image should be a Lebesgue measurable set. 01:13:53.050 --> 01:13:55.020 align:middle line:90% We'll stop there. 01:13:55.020 --> 01:14:02.000 align:middle line:90%