WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:03.936 align:middle line:90% [RUSTLING] 00:00:03.936 --> 00:00:05.412 align:middle line:90% [CLICKING] 00:00:05.412 --> 00:00:19.560 align:middle line:90% 00:00:19.560 --> 00:00:22.350 align:middle line:84% PROFESSOR: So let's finish our discussion of the Hahn-Banach 00:00:22.350 --> 00:00:23.010 align:middle line:90% Theorem. 00:00:23.010 --> 00:00:24.060 align:middle line:90% Let me just recall. 00:00:24.060 --> 00:00:30.060 align:middle line:90% 00:00:30.060 --> 00:00:42.420 align:middle line:84% This was the theorem that stated, that if v is norm space, 00:00:42.420 --> 00:00:44.130 align:middle line:90% m is a vector-- 00:00:44.130 --> 00:00:52.680 align:middle line:90% or is a subspace of v. 00:00:52.680 --> 00:00:59.745 align:middle line:84% And u is a bounded linear functional on m. 00:00:59.745 --> 00:01:07.290 align:middle line:90% 00:01:07.290 --> 00:01:12.420 align:middle line:84% So it's a linear map that also satisfies for all t and m, u 00:01:12.420 --> 00:01:15.720 align:middle line:90% of t, and absolute value is-- 00:01:15.720 --> 00:01:17.940 align:middle line:84% or the modulus of u of t is less than 00:01:17.940 --> 00:01:21.225 align:middle line:84% or equal to a constant times norm of t. 00:01:21.225 --> 00:01:25.850 align:middle line:90% 00:01:25.850 --> 00:01:31.190 align:middle line:84% Then there exists a bounded linear functional 00:01:31.190 --> 00:01:35.540 align:middle line:84% on the whole space, which extends it. 00:01:35.540 --> 00:01:37.520 align:middle line:84% So that there exists a capital U, 00:01:37.520 --> 00:01:42.980 align:middle line:84% so that when U is restricted to m this gives me little u. 00:01:42.980 --> 00:01:45.860 align:middle line:84% And it's extended continuously in the sense 00:01:45.860 --> 00:01:49.355 align:middle line:84% that it satisfies the same bound that little u does. 00:01:49.355 --> 00:01:53.920 align:middle line:90% 00:01:53.920 --> 00:02:01.540 align:middle line:84% Now for all t and v. u of t is less than 00:02:01.540 --> 00:02:05.915 align:middle line:84% or equal to a constant times the norm of t. 00:02:05.915 --> 00:02:07.540 align:middle line:84% So last time, we went through the proof 00:02:07.540 --> 00:02:14.050 align:middle line:84% of this using Zorn's lemma and the lemma 00:02:14.050 --> 00:02:17.060 align:middle line:84% that we ran up against at the end of class. 00:02:17.060 --> 00:02:20.170 align:middle line:84% Which, when writing my notes for the lecture, 00:02:20.170 --> 00:02:24.670 align:middle line:84% I got a little bit flippant towards the end 00:02:24.670 --> 00:02:26.625 align:middle line:90% and made a little small error. 00:02:26.625 --> 00:02:28.000 align:middle line:84% Which can be remedied if you just 00:02:28.000 --> 00:02:31.092 align:middle line:84% look at the lecture notes, which I have online. 00:02:31.092 --> 00:02:32.800 align:middle line:84% Or if you look at Richard's lecture notes 00:02:32.800 --> 00:02:36.220 align:middle line:84% just for the very end of concluding that Lemma we needed. 00:02:36.220 --> 00:02:38.730 align:middle line:90% 00:02:38.730 --> 00:02:43.160 align:middle line:84% So minus epsilon, we proved the Hahn-Banach Theorem. 00:02:43.160 --> 00:02:46.930 align:middle line:84% You can fill in the epsilon by just looking at the notes. 00:02:46.930 --> 00:02:52.550 align:middle line:84% But the proof of the Hahn-Banach Theorem is not the point. 00:02:52.550 --> 00:02:54.895 align:middle line:90% It's not the main point. 00:02:54.895 --> 00:02:57.380 align:middle line:84% It's important to know it, or at least have seen it, 00:02:57.380 --> 00:03:01.150 align:middle line:84% but what is important is it as a tool. 00:03:01.150 --> 00:03:08.880 align:middle line:90% 00:03:08.880 --> 00:03:11.070 align:middle line:84% And I mentioned last time, one application 00:03:11.070 --> 00:03:14.010 align:middle line:84% is to show that the dual of little l 00:03:14.010 --> 00:03:16.620 align:middle line:90% infinity is not little l1. 00:03:16.620 --> 00:03:19.800 align:middle line:84% Another application is the following is, 00:03:19.800 --> 00:03:22.110 align:middle line:84% so v is going to be a norm space. 00:03:22.110 --> 00:03:32.870 align:middle line:90% 00:03:32.870 --> 00:03:38.495 align:middle line:84% Then for all v and t, take away 0. 00:03:38.495 --> 00:03:41.660 align:middle line:90% 00:03:41.660 --> 00:03:45.170 align:middle line:84% There exists a bounded linear functional 00:03:45.170 --> 00:03:51.410 align:middle line:84% and element of the dual space such that the norm of f 00:03:51.410 --> 00:03:54.410 align:middle line:90% equals 1. 00:03:54.410 --> 00:04:07.050 align:middle line:84% And f of v gives me the norm of v. 00:04:07.050 --> 00:04:09.970 align:middle line:84% So there is a bounded linear functional 00:04:09.970 --> 00:04:17.339 align:middle line:84% that, when you apply it to v, you get the norm of v. 00:04:17.339 --> 00:04:20.250 align:middle line:84% And we'll use this just in a second to say something 00:04:20.250 --> 00:04:23.940 align:middle line:84% a little bit deeper about the relation between the dual 00:04:23.940 --> 00:04:25.810 align:middle line:84% and then taking the dual of that. 00:04:25.810 --> 00:04:28.290 align:middle line:84% But let's prove this simple theorem. 00:04:28.290 --> 00:04:29.415 align:middle line:90% So first, I define. 00:04:29.415 --> 00:04:31.940 align:middle line:90% 00:04:31.940 --> 00:04:35.380 align:middle line:84% So I have a little v that's non-0 in the space, 00:04:35.380 --> 00:04:39.790 align:middle line:84% and I define a bounded linear-- or a linear map 00:04:39.790 --> 00:04:53.300 align:middle line:84% from the span of v to c by u of lambda v 00:04:53.300 --> 00:05:01.210 align:middle line:84% equals lambda times the norm of v. 00:05:01.210 --> 00:05:04.720 align:middle line:84% Now, every element that's a scalar multiple of v 00:05:04.720 --> 00:05:06.070 align:middle line:90% can be written uniquely as this. 00:05:06.070 --> 00:05:07.450 align:middle line:84% So this function is well-defined, 00:05:07.450 --> 00:05:10.720 align:middle line:84% and it's clear that it's linear because linearity 00:05:10.720 --> 00:05:14.250 align:middle line:90% is occurring in lambda. 00:05:14.250 --> 00:05:15.190 align:middle line:90% v is fixed. 00:05:15.190 --> 00:05:15.690 align:middle line:90% Remember? 00:05:15.690 --> 00:05:20.930 align:middle line:90% 00:05:20.930 --> 00:05:29.990 align:middle line:84% So, in particular, I get that u of t 00:05:29.990 --> 00:05:31.550 align:middle line:90% is less than or equal to u-- 00:05:31.550 --> 00:05:40.660 align:middle line:84% to the norm of t for all t in the span of this fixed vector v. 00:05:40.660 --> 00:05:48.190 align:middle line:84% And you have one-- or I should say, u of v, which, remember, 00:05:48.190 --> 00:05:49.090 align:middle line:90% is just a-- 00:05:49.090 --> 00:05:51.870 align:middle line:90% 00:05:51.870 --> 00:05:56.840 align:middle line:84% So this is equal to just 1 times the norm of v. 00:05:56.840 --> 00:05:59.500 align:middle line:90% It gives me norm of v. 00:05:59.500 --> 00:06:05.970 align:middle line:84% And so, then, by the Hahn-Banach Theorem, 00:06:05.970 --> 00:06:15.450 align:middle line:84% there exists an f element of the dual extending u. 00:06:15.450 --> 00:06:24.730 align:middle line:84% Such that now, for all t and v, f of t and modulus 00:06:24.730 --> 00:06:29.920 align:middle line:84% is less than or equal to the norm of t. 00:06:29.920 --> 00:06:31.390 align:middle line:90% And in particular. 00:06:31.390 --> 00:06:36.970 align:middle line:90% 00:06:36.970 --> 00:06:52.630 align:middle line:84% So since it extends, u must have equals the norm of v. 00:06:52.630 --> 00:06:56.170 align:middle line:84% And by that inequality over there, the norm of little f 00:06:56.170 --> 00:06:57.595 align:middle line:90% must be less than or equal to 1. 00:06:57.595 --> 00:07:04.200 align:middle line:90% 00:07:04.200 --> 00:07:06.880 align:middle line:84% Well, first, let me finish with what this inequality tells you. 00:07:06.880 --> 00:07:07.755 align:middle line:90% This tells you that-- 00:07:07.755 --> 00:07:11.840 align:middle line:90% 00:07:11.840 --> 00:07:26.430 align:middle line:84% So since f of t is less than or equal to t for all t and v, 00:07:26.430 --> 00:07:29.790 align:middle line:84% this implies that the norm of f-- 00:07:29.790 --> 00:07:35.640 align:middle line:84% Which, remember, is the sup of f of t with t unit length. 00:07:35.640 --> 00:07:37.620 align:middle line:84% It must be less than or equal to 1. 00:07:37.620 --> 00:07:46.370 align:middle line:90% 00:07:46.370 --> 00:07:49.970 align:middle line:84% But I have that 1 is equal to f of b 00:07:49.970 --> 00:07:53.270 align:middle line:84% over norm of b, which must be less than 00:07:53.270 --> 00:07:55.400 align:middle line:90% or equal to the norm of f. 00:07:55.400 --> 00:08:00.530 align:middle line:84% And, therefore, the norm of f equals 1. 00:08:00.530 --> 00:08:08.170 align:middle line:84% So f is this desired, continuous linear functional on v 00:08:08.170 --> 00:08:10.750 align:middle line:84% that, when it hits v, little v, it 00:08:10.750 --> 00:08:22.790 align:middle line:90% spits out the norm of little v. 00:08:22.790 --> 00:08:27.080 align:middle line:90% The double dual of a space is-- 00:08:27.080 --> 00:08:28.580 align:middle line:84% if you can take the dual of a space, 00:08:28.580 --> 00:08:30.090 align:middle line:84% you can take the dual of that space. 00:08:30.090 --> 00:08:40.530 align:middle line:90% 00:08:40.530 --> 00:08:47.900 align:middle line:84% The double dual of a norm space v 00:08:47.900 --> 00:08:51.810 align:middle line:84% is, by definition, we denote it by v double prime. 00:08:51.810 --> 00:09:01.040 align:middle line:84% This is the dual of the dual of v. So, remember, 00:09:01.040 --> 00:09:06.740 align:middle line:84% the prime is the space of bounded linear functionals on v. 00:09:06.740 --> 00:09:10.610 align:middle line:84% And so, then, v double prime is the space 00:09:10.610 --> 00:09:13.220 align:middle line:84% of bounded linear functionals on the space 00:09:13.220 --> 00:09:24.100 align:middle line:84% of bounded linear functionals of v. 00:09:24.100 --> 00:09:27.610 align:middle line:84% Now, let me just give you an example of-- 00:09:27.610 --> 00:09:34.820 align:middle line:84% so, a quick little example of an element that you can 00:09:34.820 --> 00:09:36.695 align:middle line:90% associate in v double prime. 00:09:36.695 --> 00:09:39.830 align:middle line:90% 00:09:39.830 --> 00:09:47.860 align:middle line:84% Fix an element in capital V and define. 00:09:47.860 --> 00:09:54.880 align:middle line:90% Now, let's call it something. 00:09:54.880 --> 00:10:00.630 align:middle line:84% Let's call it t of v. So this will 00:10:00.630 --> 00:10:04.320 align:middle line:84% be an element of the dual space of v prime in the end. 00:10:04.320 --> 00:10:10.020 align:middle line:90% But t sub v by-- 00:10:10.020 --> 00:10:18.390 align:middle line:84% So this thing has to eat an element of the dull space 00:10:18.390 --> 00:10:19.680 align:middle line:90% and spit out a number. 00:10:19.680 --> 00:10:23.980 align:middle line:84% Now, think about the two pieces of data I have here. 00:10:23.980 --> 00:10:27.010 align:middle line:84% I have some fixed little v and capital V, 00:10:27.010 --> 00:10:29.980 align:middle line:84% and I would like to use that to define 00:10:29.980 --> 00:10:35.370 align:middle line:84% some bounded linear functional on capital V prime. 00:10:35.370 --> 00:10:40.170 align:middle line:84% Now, capital V prime eats elements of capital V. 00:10:40.170 --> 00:10:49.290 align:middle line:84% So I can define a map from V prime to C by that. 00:10:49.290 --> 00:10:55.880 align:middle line:90% 00:10:55.880 --> 00:11:01.430 align:middle line:84% Then I claim that T of V is an element of the double dual. 00:11:01.430 --> 00:11:05.740 align:middle line:90% 00:11:05.740 --> 00:11:08.590 align:middle line:90% So the dual space of V prime. 00:11:08.590 --> 00:11:09.740 align:middle line:90% And why is that? 00:11:09.740 --> 00:11:13.360 align:middle line:84% So first off, it has to be linear in the argument. 00:11:13.360 --> 00:11:17.590 align:middle line:84% So little v prime is in capital V prime. 00:11:17.590 --> 00:11:20.530 align:middle line:84% It has to be linear in the argument V prime. 00:11:20.530 --> 00:11:21.640 align:middle line:90% And this is clear. 00:11:21.640 --> 00:11:23.600 align:middle line:90% So remember, little v is fixed. 00:11:23.600 --> 00:11:27.580 align:middle line:84% So if I take a linear combination 00:11:27.580 --> 00:11:29.650 align:middle line:84% of two elements of capital V prime, 00:11:29.650 --> 00:11:34.430 align:middle line:84% then this expression is clearly linear in V prime. 00:11:34.430 --> 00:11:40.590 align:middle line:90% So T sub V is linear clearly. 00:11:40.590 --> 00:11:42.870 align:middle line:84% Now we just need to check that it's bounded. 00:11:42.870 --> 00:11:46.290 align:middle line:90% And if I take-- 00:11:46.290 --> 00:11:54.280 align:middle line:90% so T sub v is linear. 00:11:54.280 --> 00:11:56.620 align:middle line:90% This is a check. 00:11:56.620 --> 00:11:59.190 align:middle line:90% Just how we talk through it. 00:11:59.190 --> 00:12:11.460 align:middle line:84% And T sub v is bounded because if we take T sub v of v prime, 00:12:11.460 --> 00:12:15.150 align:middle line:84% if we take its modulus, this is by definition 00:12:15.150 --> 00:12:17.130 align:middle line:90% equal to v prime of v. 00:12:17.130 --> 00:12:21.240 align:middle line:84% Now, v prime is a bounded linear functional on v. 00:12:21.240 --> 00:12:25.890 align:middle line:84% So this is less than or equal to the norm of v prime times 00:12:25.890 --> 00:12:33.350 align:middle line:84% the norm of v. And in fact, let me write it this way. 00:12:33.350 --> 00:12:35.100 align:middle line:90% Norm of v, v prime. 00:12:35.100 --> 00:12:41.150 align:middle line:84% So we've shown that this linear operator from v 00:12:41.150 --> 00:12:46.550 align:middle line:84% prime to C in modulus is bounded by this constant times 00:12:46.550 --> 00:12:48.810 align:middle line:90% the norm of v prime. 00:12:48.810 --> 00:12:52.070 align:middle line:84% And therefore, we conclude that T of v-- 00:12:52.070 --> 00:13:00.200 align:middle line:84% or T sub v is in the dual of V prime, the double dual. 00:13:00.200 --> 00:13:08.780 align:middle line:84% And the norm of this as an operator going from v prime 00:13:08.780 --> 00:13:13.250 align:middle line:84% to C, which we've just shown, is bounded by this constant. 00:13:13.250 --> 00:13:16.370 align:middle line:84% Remember, the operator norm is the best constant 00:13:16.370 --> 00:13:17.580 align:middle line:90% that appears here. 00:13:17.580 --> 00:13:23.030 align:middle line:84% And therefore, this is less than or equal to that. 00:13:23.030 --> 00:13:28.100 align:middle line:90% 00:13:28.100 --> 00:13:33.050 align:middle line:84% So we've shown that every element of v, 00:13:33.050 --> 00:13:39.110 align:middle line:84% we can associate an element of the double dual, the dual space 00:13:39.110 --> 00:13:45.150 align:middle line:84% of the dual space, via this relation. 00:13:45.150 --> 00:13:48.620 align:middle line:84% So an element of v can be defined 00:13:48.620 --> 00:13:52.160 align:middle line:84% as an element of v double prime by letting 00:13:52.160 --> 00:13:56.360 align:middle line:84% it act on v prime by this formula here. 00:13:56.360 --> 00:14:06.940 align:middle line:90% 00:14:06.940 --> 00:14:08.530 align:middle line:84% But we can say a little bit more. 00:14:08.530 --> 00:14:13.330 align:middle line:84% And let me just introduce some terminology. 00:14:13.330 --> 00:14:26.430 align:middle line:84% Let v be in capital V. Or, no, I'm not quite there. 00:14:26.430 --> 00:14:30.330 align:middle line:90% 00:14:30.330 --> 00:14:43.460 align:middle line:84% If V and W are normed spaces, then we 00:14:43.460 --> 00:14:49.790 align:middle line:84% say that a bounded linear operator from V to W 00:14:49.790 --> 00:14:54.900 align:middle line:90% is isometric. 00:14:54.900 --> 00:14:57.470 align:middle line:84% So isometric meaning doesn't change length, 00:14:57.470 --> 00:15:00.800 align:middle line:90% doesn't change distances. 00:15:00.800 --> 00:15:09.860 align:middle line:84% If for all v and capital V the length of T of V 00:15:09.860 --> 00:15:18.920 align:middle line:90% is equal to the length of V. 00:15:18.920 --> 00:15:26.270 align:middle line:84% And now, the next theorem is that this map, this relation 00:15:26.270 --> 00:15:29.630 align:middle line:84% I've given between elements of v and the double dual 00:15:29.630 --> 00:15:32.950 align:middle line:90% is, in fact isometric. 00:15:32.950 --> 00:15:39.460 align:middle line:90% 00:15:39.460 --> 00:15:54.350 align:middle line:84% So let v be in V. And as before, define T sub v as the map from v 00:15:54.350 --> 00:16:03.930 align:middle line:84% prime to C via T sub v needs to be elements of the dual. 00:16:03.930 --> 00:16:08.270 align:middle line:84% So you can spit out numbers so it's defined in this way. 00:16:08.270 --> 00:16:14.930 align:middle line:90% 00:16:14.930 --> 00:16:25.440 align:middle line:84% Then the map T from V to V double prime, 00:16:25.440 --> 00:16:36.880 align:middle line:84% where this map T is taking V to T little v is isometric. 00:16:36.880 --> 00:16:41.910 align:middle line:90% 00:16:41.910 --> 00:16:46.460 align:middle line:84% So it's a bounded linear operator from V 00:16:46.460 --> 00:16:47.780 align:middle line:90% to its double dual. 00:16:47.780 --> 00:16:52.070 align:middle line:84% And the length of a vector in v is 00:16:52.070 --> 00:16:56.180 align:middle line:84% same as the length of its image in the double dual. 00:16:56.180 --> 00:16:59.510 align:middle line:84% Which is the length in the double dual 00:16:59.510 --> 00:17:01.415 align:middle line:84% is defined in terms of the operator norm. 00:17:01.415 --> 00:17:04.960 align:middle line:90% 00:17:04.960 --> 00:17:10.780 align:middle line:84% So we've done basically all of the work when 00:17:10.780 --> 00:17:12.260 align:middle line:90% I was discussing the example. 00:17:12.260 --> 00:17:33.740 align:middle line:84% So we've shown already that the map v to T of v, 00:17:33.740 --> 00:17:40.010 align:middle line:84% this is a bounded linear operator from v 00:17:40.010 --> 00:17:41.100 align:middle line:90% to v double prime. 00:17:41.100 --> 00:17:43.400 align:middle line:84% So we showed it's bounded, namely 00:17:43.400 --> 00:17:45.290 align:middle line:90% that the image is bounded by-- 00:17:45.290 --> 00:17:51.680 align:middle line:84% or the norm of the image is bounded by the norm of v. 00:17:51.680 --> 00:17:54.470 align:middle line:84% But it's also clear that this is linear in little v. 00:17:54.470 --> 00:17:57.320 align:middle line:84% So a minute ago, we showed it's linear in v prime. 00:17:57.320 --> 00:18:02.000 align:middle line:84% But it's also linear in v. Because for each fixed v prime, 00:18:02.000 --> 00:18:07.280 align:middle line:84% this is linear in v. So this is a bounded-- 00:18:07.280 --> 00:18:12.140 align:middle line:84% this map from v taking you to this element in T of v, 00:18:12.140 --> 00:18:15.590 align:middle line:84% this is a bounded linear operator from v 00:18:15.590 --> 00:18:18.180 align:middle line:90% to v double prime. 00:18:18.180 --> 00:18:21.830 align:middle line:84% So what's left is to show that it's isometric. 00:18:21.830 --> 00:18:25.500 align:middle line:90% 00:18:25.500 --> 00:18:36.460 align:middle line:84% So we've shown this and that the norm of the image in the double 00:18:36.460 --> 00:18:40.510 align:middle line:84% dual is less than or equal to the norm of v. 00:18:40.510 --> 00:18:43.630 align:middle line:84% Now to show it's isometric, we must show that the norm of v 00:18:43.630 --> 00:18:45.340 align:middle line:90% equals T of v. 00:18:45.340 --> 00:18:55.020 align:middle line:84% So by the theorem we just proved a minute ago, 00:18:55.020 --> 00:19:13.780 align:middle line:84% so first off, so as in the statement of the theorem, 00:19:13.780 --> 00:19:20.100 align:middle line:84% let me denote this map by this is T. 00:19:20.100 --> 00:19:27.540 align:middle line:84% So we've shown that the norm of T is less than or equal to 1. 00:19:27.540 --> 00:19:31.140 align:middle line:84% Because we've shown that the norm of the image 00:19:31.140 --> 00:19:35.730 align:middle line:84% is always less than or equal to the norm of the input. 00:19:35.730 --> 00:19:38.100 align:middle line:84% And now to show that it's isometric we just need 00:19:38.100 --> 00:19:39.585 align:middle line:90% to show the norm of T equals 1. 00:19:39.585 --> 00:19:50.970 align:middle line:90% 00:19:50.970 --> 00:19:56.760 align:middle line:90% So let v be in v non-zero. 00:19:56.760 --> 00:20:03.370 align:middle line:90% 00:20:03.370 --> 00:20:05.005 align:middle line:90% with norm v equal to 1. 00:20:05.005 --> 00:20:21.630 align:middle line:90% 00:20:21.630 --> 00:20:26.820 align:middle line:84% Well, so I just said something really stupid a minute ago. 00:20:26.820 --> 00:20:31.420 align:middle line:84% Let me go back to what I had written down there a minute ago. 00:20:31.420 --> 00:20:36.030 align:middle line:84% So not that-- so remember, we're trying 00:20:36.030 --> 00:20:40.290 align:middle line:84% to show not that the norm is one but that the norm of the image 00:20:40.290 --> 00:20:42.920 align:middle line:84% is equal to the norm of the input. 00:20:42.920 --> 00:20:47.460 align:middle line:84% I got backwards there for a minute, sorry. 00:20:47.460 --> 00:20:52.800 align:middle line:84% And the norm of the image is less than 00:20:52.800 --> 00:20:55.217 align:middle line:84% or equal to the norm of the input. 00:20:55.217 --> 00:20:57.300 align:middle line:84% So now we just want to show the reverse inequality 00:20:57.300 --> 00:21:12.490 align:middle line:84% for all little v in capital V. Now we show all v in capital V, 00:21:12.490 --> 00:21:18.908 align:middle line:84% T of v is equal to the length of v. 00:21:18.908 --> 00:21:20.700 align:middle line:84% And to do that, we use this theorem we just 00:21:20.700 --> 00:21:22.330 align:middle line:90% proved a minute ago. 00:21:22.330 --> 00:21:26.250 align:middle line:90% So this is clear if v is 0. 00:21:26.250 --> 00:21:33.460 align:middle line:84% So suppose v is non-zero, then there 00:21:33.460 --> 00:21:40.900 align:middle line:84% exists an element of the dual space by the previous theorem 00:21:40.900 --> 00:21:55.960 align:middle line:84% such that the norm of f equals 1 and f of v equals the norm of v. 00:21:55.960 --> 00:22:03.270 align:middle line:84% Then the norm of v is equal to f of v. 00:22:03.270 --> 00:22:07.970 align:middle line:84% And I can even put modulus on that. 00:22:07.970 --> 00:22:10.430 align:middle line:84% And this is less than or equal to 00:22:10.430 --> 00:22:12.740 align:middle line:90% though, thinking of this as a-- 00:22:12.740 --> 00:22:23.390 align:middle line:90% 00:22:23.390 --> 00:22:25.970 align:middle line:84% so now thinking of this as an operator 00:22:25.970 --> 00:22:29.990 align:middle line:84% from bounded linear functional, so [INAUDIBLE], 00:22:29.990 --> 00:22:32.090 align:middle line:84% so as an element of the dual this is less than 00:22:32.090 --> 00:22:38.480 align:middle line:84% or equal to the norm of T sub v times the norm of f. 00:22:38.480 --> 00:22:46.109 align:middle line:90% 00:22:46.109 --> 00:22:48.480 align:middle line:90% And now the norm of f is 1. 00:22:48.480 --> 00:22:54.000 align:middle line:84% So this is equal to the norm of T sub v. And therefore, 00:22:54.000 --> 00:22:58.560 align:middle line:84% v, the norm of v, is less than or equal to the norm of T sub v. 00:22:58.560 --> 00:23:02.610 align:middle line:84% And since I already had the reverse inequality, 00:23:02.610 --> 00:23:11.980 align:middle line:84% I conclude that the norm of T sub v equals the norm of v. 00:23:11.980 --> 00:23:20.770 align:middle line:84% And thus, this map going from v to v, to the double dual of v 00:23:20.770 --> 00:23:23.860 align:middle line:84% described this way is, in fact, isometric. 00:23:23.860 --> 00:23:26.695 align:middle line:84% It's a bounded linear operator that preserves links. 00:23:26.695 --> 00:23:30.990 align:middle line:90% 00:23:30.990 --> 00:23:32.940 align:middle line:90% Now we have a special name. 00:23:32.940 --> 00:23:38.790 align:middle line:84% So first off, it should be clear that isometric bounded linear 00:23:38.790 --> 00:23:41.340 align:middle line:90% operators are one to one. 00:23:41.340 --> 00:23:44.340 align:middle line:84% So for something to be one to one for a linear operator, that 00:23:44.340 --> 00:23:47.520 align:middle line:84% means the only thing that gets sent to 0 is 0. 00:23:47.520 --> 00:23:53.820 align:middle line:84% And from this equality here we have that if this equals 0, 00:23:53.820 --> 00:23:56.100 align:middle line:90% then the vector had to be 0. 00:23:56.100 --> 00:24:02.860 align:middle line:84% So isometric bounded linear operators are always one to one. 00:24:02.860 --> 00:24:05.400 align:middle line:84% So what this theorem tells you is 00:24:05.400 --> 00:24:09.210 align:middle line:84% that this map defined in this way 00:24:09.210 --> 00:24:15.480 align:middle line:84% gives you a isometric injection from v to v double prime, 00:24:15.480 --> 00:24:18.790 align:middle line:90% from v to its double dual. 00:24:18.790 --> 00:24:24.150 align:middle line:84% So I have this map that goes into the double dual that's 00:24:24.150 --> 00:24:29.630 align:middle line:84% isometric, meaning it doesn't change distances. 00:24:29.630 --> 00:24:30.800 align:middle line:90% Is it onto? 00:24:30.800 --> 00:24:35.540 align:middle line:84% Is the double dual always equal to the original space itself? 00:24:35.540 --> 00:24:38.750 align:middle line:84% So we have a name for spaces that satisfy that. 00:24:38.750 --> 00:24:43.360 align:middle line:90% 00:24:43.360 --> 00:24:59.780 align:middle line:84% So a Banach space V is reflexive if using-- 00:24:59.780 --> 00:25:00.980 align:middle line:90% if this map is onto. 00:25:00.980 --> 00:25:04.320 align:middle line:90% 00:25:04.320 --> 00:25:12.010 align:middle line:84% If V equals v double prime, in the sense 00:25:12.010 --> 00:25:18.100 align:middle line:84% that this map taking element of capital V 00:25:18.100 --> 00:25:25.570 align:middle line:84% to an element of the double dual that, as we defined earlier, 00:25:25.570 --> 00:25:28.210 align:middle line:90% is onto. 00:25:28.210 --> 00:25:33.270 align:middle line:90% 00:25:33.270 --> 00:25:43.590 align:middle line:84% Now for example, and you can check that. 00:25:43.590 --> 00:25:45.750 align:middle line:84% This may seem like abstract stuff. 00:25:45.750 --> 00:25:51.060 align:middle line:84% But for little lp spaces, this letting 00:25:51.060 --> 00:25:56.100 align:middle line:84% the dual space eat a vector or whatever, remember, 00:25:56.100 --> 00:25:58.800 align:middle line:84% we identify the dual space of little lp 00:25:58.800 --> 00:26:03.900 align:middle line:84% with little lq, where 1 over p plus 1 over q equals 1. 00:26:03.900 --> 00:26:16.070 align:middle line:84% So little lp is reflexive for p between 1 and infinity 00:26:16.070 --> 00:26:20.000 align:middle line:84% because the dual of little lp is going 00:26:20.000 --> 00:26:24.110 align:middle line:84% to be little lq, where 1 over p plus 1 over q equals 1. 00:26:24.110 --> 00:26:26.930 align:middle line:90% And the dual of 1 over-- 00:26:26.930 --> 00:26:30.290 align:middle line:84% of little lq is going to be a little lp as long as q 00:26:30.290 --> 00:26:32.570 align:middle line:90% is between 1 and infinity. 00:26:32.570 --> 00:26:36.560 align:middle line:84% There is that one case where the dual is not given 00:26:36.560 --> 00:26:39.610 align:middle line:90% by what you think it should be. 00:26:39.610 --> 00:26:44.370 align:middle line:90% So little l1 is not reflexive. 00:26:44.370 --> 00:26:48.610 align:middle line:90% 00:26:48.610 --> 00:26:55.840 align:middle line:84% Since if I take a look at the dual of little l1, 00:26:55.840 --> 00:27:04.450 align:middle line:84% so this should be 1, this is equal to l infinity. 00:27:04.450 --> 00:27:09.280 align:middle line:84% And as you'll show in the homework, the dual of little l 00:27:09.280 --> 00:27:12.670 align:middle line:84% infinity is not equal to little l1. 00:27:12.670 --> 00:27:14.800 align:middle line:84% So the dual of the dual is not going 00:27:14.800 --> 00:27:18.115 align:middle line:84% to give you back the space you started with. 00:27:18.115 --> 00:27:21.890 align:middle line:90% 00:27:21.890 --> 00:27:24.800 align:middle line:84% And I don't know if I'm going to put this in the assignment 00:27:24.800 --> 00:27:28.730 align:middle line:90% or eventually on a midterm. 00:27:28.730 --> 00:27:30.800 align:middle line:84% The space C0, which was the space 00:27:30.800 --> 00:27:34.460 align:middle line:84% of all sequences converging to 0, this is not reflexive either. 00:27:34.460 --> 00:27:39.490 align:middle line:90% 00:27:39.490 --> 00:27:46.000 align:middle line:84% The reason is because you can, in fact, identify the dual of C0 00:27:46.000 --> 00:27:47.062 align:middle line:90% with little l1. 00:27:47.062 --> 00:27:49.590 align:middle line:90% 00:27:49.590 --> 00:27:58.740 align:middle line:84% And the dual of little l1 is little L infinity. 00:27:58.740 --> 00:28:01.110 align:middle line:90% Which does not equal C0. 00:28:01.110 --> 00:28:04.910 align:middle line:90% 00:28:04.910 --> 00:28:09.590 align:middle line:84% So this also, you can see how the space is 00:28:09.590 --> 00:28:13.550 align:middle line:90% a subspace of the double dual. 00:28:13.550 --> 00:28:17.300 align:middle line:84% The space of all sequences converging 00:28:17.300 --> 00:28:23.330 align:middle line:84% to 0, that's a subspace of the space of all bounded sequences. 00:28:23.330 --> 00:28:30.970 align:middle line:84% So you see this, I shouldn't say by hand, but OK, by hand, 00:28:30.970 --> 00:28:33.430 align:middle line:90% that the original space-- 00:28:33.430 --> 00:28:37.150 align:middle line:84% forget this prime-- is a subspace of the double dual 00:28:37.150 --> 00:28:39.280 align:middle line:84% which in this case is little infinity. 00:28:39.280 --> 00:28:48.500 align:middle line:90% 00:28:48.500 --> 00:28:56.900 align:middle line:84% So that concludes the general stuff 00:28:56.900 --> 00:29:00.040 align:middle line:90% about Banach spaces for now. 00:29:00.040 --> 00:29:01.540 align:middle line:84% And we're going to turn to Lebesgue 00:29:01.540 --> 00:29:04.420 align:middle line:90% measure and integration. 00:29:04.420 --> 00:29:09.600 align:middle line:84% Because so we've been talking about little lp spaces. 00:29:09.600 --> 00:29:11.320 align:middle line:90% These are spaces of sequences. 00:29:11.320 --> 00:29:15.570 align:middle line:84% And so you might think, well, maybe we can define big LP 00:29:15.570 --> 00:29:20.460 align:middle line:84% spaces to be, let's say, Riemann integral functions whose 00:29:20.460 --> 00:29:22.950 align:middle line:84% p'th power is integrable or something like that. 00:29:22.950 --> 00:29:28.240 align:middle line:84% Just how these are p'th power summable. 00:29:28.240 --> 00:29:42.420 align:middle line:84% So moving on now to Lebesgue measure and integration. 00:29:42.420 --> 00:29:50.020 align:middle line:90% 00:29:50.020 --> 00:29:51.220 align:middle line:90% Missing a letter there. 00:29:51.220 --> 00:29:55.760 align:middle line:90% 00:29:55.760 --> 00:29:58.280 align:middle line:84% Now, why the Lebesgue measure and integration, 00:29:58.280 --> 00:30:02.120 align:middle line:84% why not just stick to Riemann integration? 00:30:02.120 --> 00:30:04.080 align:middle line:90% There's a couple of reasons. 00:30:04.080 --> 00:30:08.210 align:middle line:84% One is Lebesgue integration has much 00:30:08.210 --> 00:30:09.770 align:middle line:90% better convergence theorems. 00:30:09.770 --> 00:30:14.330 align:middle line:84% So you only have really one convergence theorem 00:30:14.330 --> 00:30:19.040 align:middle line:84% for Riemann integration, which is the uniform limit of Riemann 00:30:19.040 --> 00:30:21.110 align:middle line:84% integrable functions is Riemann integrable. 00:30:21.110 --> 00:30:25.850 align:middle line:84% And the integral of the limits is the limit of the integrals. 00:30:25.850 --> 00:30:29.850 align:middle line:90% 00:30:29.850 --> 00:30:33.240 align:middle line:84% But there are much better limit theorems. 00:30:33.240 --> 00:30:36.270 align:middle line:84% And in the sense that they're more useful, 00:30:36.270 --> 00:30:41.580 align:middle line:84% you can use them more often, and therefore prove better things. 00:30:41.580 --> 00:30:47.220 align:middle line:84% And maybe you're just like, OK, so what? 00:30:47.220 --> 00:30:49.860 align:middle line:84% But there's even bigger reasons why 00:30:49.860 --> 00:30:51.930 align:middle line:84% you consider Lebesgue measure and integration. 00:30:51.930 --> 00:30:54.570 align:middle line:84% And this is because that if you look 00:30:54.570 --> 00:31:00.750 align:middle line:84% at the space of Riemann integrable functions on a, b, 00:31:00.750 --> 00:31:10.140 align:middle line:84% say, 0, 1, say, let's make it concrete, then let me in fact 00:31:10.140 --> 00:31:10.815 align:middle line:90% write this down. 00:31:10.815 --> 00:31:14.130 align:middle line:90% 00:31:14.130 --> 00:31:15.705 align:middle line:84% L1-- and I'm going to put an R here. 00:31:15.705 --> 00:31:20.230 align:middle line:90% 00:31:20.230 --> 00:31:28.660 align:middle line:84% This is a set of all f from 0, 1 C such that f 00:31:28.660 --> 00:31:34.480 align:middle line:90% is Riemann integrable on 0, 1. 00:31:34.480 --> 00:31:39.670 align:middle line:84% Now in 18.100 and 100A, these are usually real-valued. 00:31:39.670 --> 00:31:42.077 align:middle line:84% But when I mean Riemann integrable and its complex 00:31:42.077 --> 00:31:44.410 align:middle line:84% valued, I just mean the real part and the imaginary part 00:31:44.410 --> 00:31:46.800 align:middle line:90% are Riemann integrable. 00:31:46.800 --> 00:31:49.440 align:middle line:90% 00:31:49.440 --> 00:31:53.230 align:middle line:84% So you don't need to know anything fancy. 00:31:53.230 --> 00:31:58.705 align:middle line:84% And if I define a norm on capital L1 by-- 00:31:58.705 --> 00:32:15.010 align:middle line:90% 00:32:15.010 --> 00:32:17.530 align:middle line:84% so I have this space I have this norm. 00:32:17.530 --> 00:32:19.090 align:middle line:90% Now, it's not quite a norm. 00:32:19.090 --> 00:32:21.280 align:middle line:84% Because you can have Riemann integrable 00:32:21.280 --> 00:32:25.780 align:middle line:84% functions which are not 0 everywhere, whose integral is 0. 00:32:25.780 --> 00:32:28.810 align:middle line:84% If you have a function that's 0 except for at one point, 00:32:28.810 --> 00:32:32.530 align:middle line:84% and at that one point it's 5, well, the function's non-zero, 00:32:32.530 --> 00:32:33.910 align:middle line:90% but the integral is 0. 00:32:33.910 --> 00:32:35.770 align:middle line:90% So this is a semi-norm. 00:32:35.770 --> 00:32:37.465 align:middle line:90% But let's imagine it's a norm. 00:32:37.465 --> 00:32:41.230 align:middle line:90% 00:32:41.230 --> 00:32:54.260 align:middle line:84% So the problem is, is that even if I mod out by those things 00:32:54.260 --> 00:32:56.780 align:middle line:84% that give me a semi-norm equal to 0 00:32:56.780 --> 00:32:59.630 align:middle line:84% so that I get an actual norm space, 00:32:59.630 --> 00:33:01.230 align:middle line:90% but ignore that for a minute. 00:33:01.230 --> 00:33:04.280 align:middle line:84% Let's just imagine that this is an actual honest to God norm. 00:33:04.280 --> 00:33:10.595 align:middle line:84% Then this thing ends up being this is not a Banach space. 00:33:10.595 --> 00:33:15.590 align:middle line:90% 00:33:15.590 --> 00:33:19.190 align:middle line:84% So one reason to consider more general integration 00:33:19.190 --> 00:33:24.960 align:middle line:84% is because Riemann integration, or at least 00:33:24.960 --> 00:33:26.820 align:middle line:84% only restricting to those functions 00:33:26.820 --> 00:33:29.040 align:middle line:84% which are Riemann integrable does not 00:33:29.040 --> 00:33:30.150 align:middle line:90% give you a Banach space. 00:33:30.150 --> 00:33:34.470 align:middle line:84% This is not-- this is a norm space modulo this little bit 00:33:34.470 --> 00:33:38.910 align:middle line:84% that there are functions that have norm 0 that aren't exactly 00:33:38.910 --> 00:33:41.070 align:middle line:90% identically 0. 00:33:41.070 --> 00:33:43.470 align:middle line:90% But it's not a Banach space. 00:33:43.470 --> 00:33:45.690 align:middle line:90% It's not complete. 00:33:45.690 --> 00:33:47.430 align:middle line:84% And in functional analysis, we're 00:33:47.430 --> 00:33:52.020 align:middle line:84% interested in spaces that are complete. 00:33:52.020 --> 00:33:56.850 align:middle line:84% And not just in functional analysis for the abstract love 00:33:56.850 --> 00:34:02.898 align:middle line:84% of it, but in problems where, as I 00:34:02.898 --> 00:34:04.440 align:middle line:84% said at the beginning of this course, 00:34:04.440 --> 00:34:07.020 align:middle line:84% where you have differentiable equations 00:34:07.020 --> 00:34:10.350 align:middle line:84% or some functional defined on some space functions 00:34:10.350 --> 00:34:11.969 align:middle line:84% you want to deal with complete spaces. 00:34:11.969 --> 00:34:21.659 align:middle line:84% And a lot of functionals are defined on not just L1 but say 00:34:21.659 --> 00:34:25.830 align:middle line:84% LP, where now I replace this by a p'th power. 00:34:25.830 --> 00:34:33.340 align:middle line:84% So in order to be able to say certain things exist or get 00:34:33.340 --> 00:34:35.530 align:middle line:84% to the heart of the subject, I need 00:34:35.530 --> 00:34:36.867 align:middle line:90% these spaces to be complete. 00:34:36.867 --> 00:34:39.159 align:middle line:84% And if I just restrict to Riemann integrable functions, 00:34:39.159 --> 00:34:40.219 align:middle line:90% it's not complete. 00:34:40.219 --> 00:34:44.560 align:middle line:84% So we have to do Lebesgue integration. 00:34:44.560 --> 00:34:47.020 align:middle line:84% Now, what we'll find out is that-- 00:34:47.020 --> 00:34:50.620 align:middle line:84% and this is how it's done in the original notes for this class, 00:34:50.620 --> 00:34:53.900 align:middle line:84% is that this is not a Banach space. 00:34:53.900 --> 00:34:56.179 align:middle line:90% But I can take its completion. 00:34:56.179 --> 00:34:59.300 align:middle line:84% Just like the set of rational numbers is not complete, 00:34:59.300 --> 00:35:01.820 align:middle line:84% I can take it's what's called completion 00:35:01.820 --> 00:35:03.560 align:middle line:90% and get the set of real numbers. 00:35:03.560 --> 00:35:06.500 align:middle line:84% I could take this space, take its completion, 00:35:06.500 --> 00:35:11.870 align:middle line:84% and what I get is a abstract space, which I can actually 00:35:11.870 --> 00:35:16.310 align:middle line:84% identify with the space of Lebesgue integrable functions. 00:35:16.310 --> 00:35:17.790 align:middle line:84% That's how it's done in the notes. 00:35:17.790 --> 00:35:20.540 align:middle line:84% But I don't want to do it that way. 00:35:20.540 --> 00:35:24.680 align:middle line:84% Instead, we're just going to build up Lebesgue measure 00:35:24.680 --> 00:35:26.910 align:middle line:84% and integration from the ground up. 00:35:26.910 --> 00:35:31.940 align:middle line:84% And we will see at one point that these functions 00:35:31.940 --> 00:35:35.240 align:middle line:84% are in fact dense in the space of Lebesgue integrable 00:35:35.240 --> 00:35:36.110 align:middle line:90% functions. 00:35:36.110 --> 00:35:43.500 align:middle line:84% And therefore, we conclude that the abstract completion-- 00:35:43.500 --> 00:35:46.190 align:middle line:84% if you haven't covered completions, that's fine-- 00:35:46.190 --> 00:35:50.300 align:middle line:84% of this space is, in fact, the integrable-- 00:35:50.300 --> 00:35:53.400 align:middle line:84% the space of Lebesgue integrable functions. 00:35:53.400 --> 00:36:01.930 align:middle line:84% So summing it up is that we have to do Lebesgue measure 00:36:01.930 --> 00:36:04.180 align:middle line:84% and integration because these spaces are not complete. 00:36:04.180 --> 00:36:06.840 align:middle line:90% 00:36:06.840 --> 00:36:10.410 align:middle line:84% And we have better convergence theorems in these spaces. 00:36:10.410 --> 00:36:14.760 align:middle line:84% But it just takes a little bit of effort to learn. 00:36:14.760 --> 00:36:17.370 align:middle line:84% Not much, but I mean, it's quite intuitive 00:36:17.370 --> 00:36:21.540 align:middle line:84% once you start seeing the arguments and going through it. 00:36:21.540 --> 00:36:27.290 align:middle line:90% 00:36:27.290 --> 00:36:30.520 align:middle line:84% So we're going to be defining a new notion of integration that's 00:36:30.520 --> 00:36:33.550 align:middle line:84% more general than Riemann integration. 00:36:33.550 --> 00:36:36.330 align:middle line:90% 00:36:36.330 --> 00:36:41.510 align:middle line:84% And so, integration should somehow 00:36:41.510 --> 00:36:46.490 align:middle line:84% be the theory of area underneath the curve. 00:36:46.490 --> 00:36:59.460 align:middle line:84% So the simplest type of functions 00:36:59.460 --> 00:37:09.990 align:middle line:84% you should consider first is, if I have a subset E, 00:37:09.990 --> 00:37:13.620 align:middle line:84% maybe it's crazy, or maybe it's simple. 00:37:13.620 --> 00:37:22.010 align:middle line:84% And I have a function, which is just a 1 when I'm on it and 0 00:37:22.010 --> 00:37:22.750 align:middle line:90% otherwise. 00:37:22.750 --> 00:37:31.400 align:middle line:90% 00:37:31.400 --> 00:37:37.390 align:middle line:84% So I'll denote this function by 1 with an E beneath it. 00:37:37.390 --> 00:37:42.890 align:middle line:84% Then in some sense, I want to be able to integrate this function. 00:37:42.890 --> 00:37:53.020 align:middle line:84% So 1E is the function that's 1 if x is in E and 0 if x is not 00:37:53.020 --> 00:37:56.110 align:middle line:84% in E. So if E is just the interval a, b, 00:37:56.110 --> 00:37:59.500 align:middle line:84% then I just have the function that 0 outside of a, b and 1 00:37:59.500 --> 00:38:00.400 align:middle line:90% over a, b. 00:38:00.400 --> 00:38:06.195 align:middle line:84% So the question is, how do we integrate such functions? 00:38:06.195 --> 00:38:11.450 align:middle line:84% Or our first task is to define the integral 00:38:11.450 --> 00:38:13.350 align:middle line:90% for these types of functions. 00:38:13.350 --> 00:38:16.490 align:middle line:84% Well, since the integral should be 00:38:16.490 --> 00:38:19.280 align:middle line:84% a theory of the area underneath the curve, 00:38:19.280 --> 00:38:22.130 align:middle line:90% for example, if E equals-- 00:38:22.130 --> 00:38:25.370 align:middle line:84% so this is all just discussion right now, I should add. 00:38:25.370 --> 00:38:26.958 align:middle line:90% So this is motivation. 00:38:26.958 --> 00:38:29.000 align:middle line:84% In a minute I'll get to definitions and theorems. 00:38:29.000 --> 00:38:32.570 align:middle line:90% But this is just a discussion. 00:38:32.570 --> 00:38:35.930 align:middle line:84% If E equals a, b so that I'm just-- 00:38:35.930 --> 00:38:51.030 align:middle line:84% I have a, b, 1, 0 outside, then the integral of this guy 00:38:51.030 --> 00:38:57.980 align:middle line:84% in whatever sense this integral is, 00:38:57.980 --> 00:39:01.950 align:middle line:84% this should be at least the area underneath this curve. 00:39:01.950 --> 00:39:05.690 align:middle line:90% So it should be b minus a. 00:39:05.690 --> 00:39:12.590 align:middle line:84% Which is the length of the interval a, b. 00:39:12.590 --> 00:39:29.540 align:middle line:84% So and therefore, if we have a general set E, 00:39:29.540 --> 00:39:37.800 align:middle line:84% we should expect that our integral over the function, 00:39:37.800 --> 00:39:41.540 align:middle line:84% which is 1 on E and 0 off it, should somehow-- 00:39:41.540 --> 00:39:45.470 align:middle line:84% this should be the length of E. But length is not a very good 00:39:45.470 --> 00:39:48.710 align:middle line:84% word because length applies to an interval a, 00:39:48.710 --> 00:39:50.930 align:middle line:84% b because there's a start and there's a stop. 00:39:50.930 --> 00:39:53.510 align:middle line:84% And everything in between is in the set. 00:39:53.510 --> 00:39:59.240 align:middle line:84% So rather than write the length of E, I'll write m of E. 00:39:59.240 --> 00:40:06.590 align:middle line:84% And m of E being the what I'll say 00:40:06.590 --> 00:40:12.800 align:middle line:84% is a measure of E. A measure of how much of E there is. 00:40:12.800 --> 00:40:16.060 align:middle line:84% And so, our first task, if we're hoping 00:40:16.060 --> 00:40:20.350 align:middle line:84% to develop a notion of integration more 00:40:20.350 --> 00:40:25.630 align:middle line:84% general than Riemann, in which the resulting spaces are 00:40:25.630 --> 00:40:28.090 align:middle line:84% a Banach space, that's really the goal in the end 00:40:28.090 --> 00:40:33.220 align:middle line:84% we should start by defining what does this mean? 00:40:33.220 --> 00:40:36.370 align:middle line:90% What is a measure of a set? 00:40:36.370 --> 00:40:38.290 align:middle line:90% Which sets are we measuring? 00:40:38.290 --> 00:40:42.700 align:middle line:90% 00:40:42.700 --> 00:40:56.785 align:middle line:84% So our task right now is define before we even 00:40:56.785 --> 00:40:59.410 align:middle line:84% get to integration, we should be able to integrate the simplest 00:40:59.410 --> 00:41:00.220 align:middle line:90% types of functions. 00:41:00.220 --> 00:41:11.550 align:middle line:84% And this requires us to be able to define the measure of subsets 00:41:11.550 --> 00:41:14.550 align:middle line:84% of R. And this is Lebesgue measure 00:41:14.550 --> 00:41:16.630 align:middle line:90% that we'll be developing. 00:41:16.630 --> 00:41:25.510 align:middle line:84% So what are some properties that we 00:41:25.510 --> 00:41:34.190 align:middle line:84% want that a reasonable measure of sets to have? 00:41:34.190 --> 00:41:36.610 align:middle line:84% So the first one we would like is, 00:41:36.610 --> 00:41:38.500 align:middle line:84% we should be able to measure everything. 00:41:38.500 --> 00:41:40.510 align:middle line:84% What's awful about Riemann integration 00:41:40.510 --> 00:41:45.310 align:middle line:84% is we can't really integrate any function. 00:41:45.310 --> 00:41:49.350 align:middle line:84% But we can't even integrate this function when E is, 00:41:49.350 --> 00:41:52.440 align:middle line:84% say, the rational numbers between 0 and 1. 00:41:52.440 --> 00:41:56.520 align:middle line:84% When I have the function which is 1 on the rationals 00:41:56.520 --> 00:42:01.120 align:middle line:84% and 0 off of that, that's not Riemann integrable. 00:42:01.120 --> 00:42:03.270 align:middle line:84% So I would like to be able to measure 00:42:03.270 --> 00:42:04.985 align:middle line:90% any subset of real numbers. 00:42:04.985 --> 00:42:11.200 align:middle line:90% 00:42:11.200 --> 00:42:16.090 align:middle line:84% So I would like to be able to define the measure 00:42:16.090 --> 00:42:18.100 align:middle line:90% of any subset of real numbers. 00:42:18.100 --> 00:42:24.910 align:middle line:84% And the second property I would like is kind of a sanity check, 00:42:24.910 --> 00:42:37.380 align:middle line:84% that if I is an interval, then the measure of E 00:42:37.380 --> 00:42:41.060 align:middle line:90% should be the length of I. 00:42:41.060 --> 00:42:44.780 align:middle line:84% And by I, I mean half open, half closed. 00:42:44.780 --> 00:42:47.150 align:middle line:84% I guess, a half open is half closed. 00:42:47.150 --> 00:42:52.060 align:middle line:84% But closed, open interval, the measure of that 00:42:52.060 --> 00:42:54.850 align:middle line:84% should not care about missing the two endpoints. 00:42:54.850 --> 00:42:57.160 align:middle line:84% And I should just get out the length of that interval 00:42:57.160 --> 00:42:58.148 align:middle line:90% b minus a. 00:42:58.148 --> 00:43:01.284 align:middle line:90% 00:43:01.284 --> 00:43:09.690 align:middle line:84% A third property is a measure of the whole 00:43:09.690 --> 00:43:14.850 align:middle line:84% should be the sum of the measures of its parts. 00:43:14.850 --> 00:43:17.370 align:middle line:84% If I have some set which can be written 00:43:17.370 --> 00:43:24.330 align:middle line:84% as a union of disjoint chunks, then the measure 00:43:24.330 --> 00:43:27.780 align:middle line:84% of that whole set should be the sum of the measure 00:43:27.780 --> 00:43:29.250 align:middle line:90% of the individual chunks. 00:43:29.250 --> 00:43:45.620 align:middle line:84% So that's stated as if En is a collection of disjoint sets. 00:43:45.620 --> 00:43:48.950 align:middle line:84% So think of these as making up a bigger set. 00:43:48.950 --> 00:43:55.750 align:middle line:84% And this is, I should say, a countable collection of disjoint 00:43:55.750 --> 00:44:03.130 align:middle line:84% sets, then I would like for the measure of their union 00:44:03.130 --> 00:44:05.740 align:middle line:90% to be the sum of the measures. 00:44:05.740 --> 00:44:08.800 align:middle line:90% 00:44:08.800 --> 00:44:12.490 align:middle line:84% This is a reasonable thing to ask for. 00:44:12.490 --> 00:44:16.270 align:middle line:84% The measure of the whole is the sum of the measure of the parts. 00:44:16.270 --> 00:44:19.270 align:middle line:84% And in the last one, which is kind of specific to how 00:44:19.270 --> 00:44:25.650 align:middle line:84% we view R is, if I take a set here next to where I'm standing, 00:44:25.650 --> 00:44:28.330 align:middle line:84% and then I take that set, don't do anything to it, 00:44:28.330 --> 00:44:31.260 align:middle line:84% I just walk it over here and take it's measure, 00:44:31.260 --> 00:44:34.200 align:middle line:84% that measure, the measure of this set which I've now 00:44:34.200 --> 00:44:35.970 align:middle line:84% walked over here should be the same 00:44:35.970 --> 00:44:40.570 align:middle line:84% as the measure of that set over when I was standing over here. 00:44:40.570 --> 00:44:45.030 align:middle line:84% So we should like m to be translation invariant. 00:44:45.030 --> 00:44:56.570 align:middle line:90% 00:44:56.570 --> 00:45:06.730 align:middle line:84% Meaning if E is in R and x is an element 00:45:06.730 --> 00:45:12.760 align:middle line:84% of r, then the measure of the set 00:45:12.760 --> 00:45:18.520 align:middle line:84% x plus E, which means the measure of the set of elements 00:45:18.520 --> 00:45:24.730 align:middle line:84% x plus y, where y is in E. So just take E and shift it by x. 00:45:24.730 --> 00:45:45.160 align:middle line:84% This should be the same as the measure of the original set E. 00:45:45.160 --> 00:45:55.160 align:middle line:84% So unfortunately, this is impossible to have 00:45:55.160 --> 00:46:04.050 align:middle line:84% a function which is defined on every subset of real numbers 00:46:04.050 --> 00:46:06.400 align:middle line:84% satisfying these three properties. 00:46:06.400 --> 00:46:11.360 align:middle line:90% So that's very unfortunate. 00:46:11.360 --> 00:46:14.910 align:middle line:90% 00:46:14.910 --> 00:46:19.770 align:middle line:84% Such a function m going from the power set of R, meaning 00:46:19.770 --> 00:46:23.730 align:middle line:84% the set of all subsets of R, and of course 00:46:23.730 --> 00:46:25.140 align:middle line:84% this thing should be non-negative 00:46:25.140 --> 00:46:36.855 align:middle line:84% since it's a measure of how much is there, does not exist. 00:46:36.855 --> 00:46:42.688 align:middle line:90% 00:46:42.688 --> 00:46:43.230 align:middle line:90% Meaning what? 00:46:43.230 --> 00:46:45.900 align:middle line:84% If you assume all of these four properties, 00:46:45.900 --> 00:46:53.150 align:middle line:84% you'll be able to come up with a set which has finite measure 00:46:53.150 --> 00:46:55.850 align:middle line:84% and then also simultaneously has infinite measure. 00:46:55.850 --> 00:46:58.700 align:middle line:90% 00:46:58.700 --> 00:47:04.650 align:middle line:84% So it's impossible to have a function which 00:47:04.650 --> 00:47:06.750 align:middle line:84% is defined for every subset of real numbers 00:47:06.750 --> 00:47:11.880 align:middle line:84% satisfying the conditions two, three, and four. 00:47:11.880 --> 00:47:14.770 align:middle line:84% It's just not logically-- it's just not possible. 00:47:14.770 --> 00:47:20.480 align:middle line:90% So that's unfortunate. 00:47:20.480 --> 00:47:29.640 align:middle line:84% But what we can do is drop the assumption 00:47:29.640 --> 00:47:35.310 align:middle line:84% that measure is defined for every subset of real numbers 00:47:35.310 --> 00:47:43.360 align:middle line:84% and focus on trying to find a set function-- 00:47:43.360 --> 00:47:46.630 align:middle line:84% I mean, defined on subsets of real numbers. 00:47:46.630 --> 00:47:48.880 align:middle line:84% So that's why I'm calling it a set function-- 00:47:48.880 --> 00:47:53.980 align:middle line:84% m which satisfies two, three and four on a big collection 00:47:53.980 --> 00:47:58.330 align:middle line:84% of sets, that's defined on a large collection of sets. 00:47:58.330 --> 00:48:02.870 align:middle line:90% 00:48:02.870 --> 00:48:07.940 align:middle line:84% And these sets for which such a measure will 00:48:07.940 --> 00:48:11.750 align:middle line:84% be defined on satisfying two, three, and four will 00:48:11.750 --> 00:48:13.910 align:middle line:90% be quite large in the end. 00:48:13.910 --> 00:48:19.130 align:middle line:84% This is the set of Lebesgue measurable sets. 00:48:19.130 --> 00:48:22.520 align:middle line:90% And m is the Lebesgue measure. 00:48:22.520 --> 00:48:26.780 align:middle line:84% And so, our goal is to construct the Lebesgue measure 00:48:26.780 --> 00:48:30.780 align:middle line:90% and Lebesgue integrable sets. 00:48:30.780 --> 00:48:32.300 align:middle line:90% So this is our task. 00:48:32.300 --> 00:48:34.980 align:middle line:90% 00:48:34.980 --> 00:48:38.730 align:middle line:84% Or I guess we had that task before. 00:48:38.730 --> 00:48:40.605 align:middle line:90% So what is the plan now? 00:48:40.605 --> 00:48:49.090 align:middle line:90% 00:48:49.090 --> 00:49:05.595 align:middle line:84% Construct an m defined for a large class-- 00:49:05.595 --> 00:49:21.340 align:middle line:90% 00:49:21.340 --> 00:49:25.300 align:middle line:84% so defined for many different sets but not necessarily-- 00:49:25.300 --> 00:49:27.130 align:middle line:90% but not every set. 00:49:27.130 --> 00:49:30.000 align:middle line:90% 00:49:30.000 --> 00:49:34.020 align:middle line:84% And these sets will be, we will call Lebesgue measurable 00:49:34.020 --> 00:49:53.210 align:middle line:84% sets such that the conditions two through four hold. 00:49:53.210 --> 00:49:56.490 align:middle line:84% That if I have an interval, the measure of that interval-- 00:49:56.490 --> 00:50:00.260 align:middle line:84% so first off, this class of sets should contain all intervals. 00:50:00.260 --> 00:50:02.360 align:middle line:84% And the measure of that interval gives me 00:50:02.360 --> 00:50:04.340 align:middle line:90% the length of the interval. 00:50:04.340 --> 00:50:08.030 align:middle line:84% And if I have a countable collection 00:50:08.030 --> 00:50:11.930 align:middle line:84% of sets for which I can measure, then their union is measurable. 00:50:11.930 --> 00:50:16.340 align:middle line:84% And the measure of that disjoint union 00:50:16.340 --> 00:50:18.020 align:middle line:84% is equal to the sum of the measures. 00:50:18.020 --> 00:50:20.600 align:middle line:84% And then it's also translation-invariant. 00:50:20.600 --> 00:50:23.330 align:middle line:90% So that's the plan. 00:50:23.330 --> 00:50:26.110 align:middle line:90% 00:50:26.110 --> 00:50:29.100 align:middle line:84% So it won't be defined on for every set. 00:50:29.100 --> 00:50:32.130 align:middle line:84% But it will be defined for a large class of sets which 00:50:32.130 --> 00:50:34.930 align:middle line:90% contain most reasonable sets. 00:50:34.930 --> 00:50:46.190 align:middle line:84% And how we're going to do this is due to Carathéodory. 00:50:46.190 --> 00:50:48.000 align:middle line:90% And we'll go about it like this. 00:50:48.000 --> 00:50:58.390 align:middle line:84% We'll first-- so how we'll construct this, 00:50:58.390 --> 00:51:05.450 align:middle line:84% we'll first construct a function m 00:51:05.450 --> 00:51:15.090 align:middle line:84% star, which is defined for every subset of real numbers. 00:51:15.090 --> 00:51:23.820 align:middle line:84% This we'll call the outer measure, which satisfies two, 00:51:23.820 --> 00:51:29.100 align:middle line:84% namely that the m star of an interval 00:51:29.100 --> 00:51:33.120 align:middle line:90% is the length of the interval 3. 00:51:33.120 --> 00:51:39.045 align:middle line:84% And I shouldn't say 3, but and almost 3. 00:51:39.045 --> 00:51:45.430 align:middle line:90% 00:51:45.430 --> 00:51:53.760 align:middle line:84% Then we restrict, I should say, satisfying 00:51:53.760 --> 00:51:59.230 align:middle line:90% two, almost three, and four. 00:51:59.230 --> 00:52:02.570 align:middle line:90% 00:52:02.570 --> 00:52:17.230 align:middle line:84% Then we restrict m star to well-behaved subsets of R. 00:52:17.230 --> 00:52:23.610 align:middle line:84% These will be the class of the big Lebesgue sets. 00:52:23.610 --> 00:52:26.715 align:middle line:84% And m will just be m star restricted to these subsets. 00:52:26.715 --> 00:52:30.590 align:middle line:90% 00:52:30.590 --> 00:52:35.300 align:middle line:84% So this is the plan of this chapter, 00:52:35.300 --> 00:52:37.550 align:middle line:90% this part of the course. 00:52:37.550 --> 00:52:42.060 align:middle line:90% Again, this is all discussion. 00:52:42.060 --> 00:52:47.340 align:middle line:84% So our first topic will be m star, 00:52:47.340 --> 00:52:48.975 align:middle line:90% which is called outer measure. 00:52:48.975 --> 00:52:58.270 align:middle line:90% 00:52:58.270 --> 00:53:02.430 align:middle line:84% So this was all game plan discussion. 00:53:02.430 --> 00:53:06.010 align:middle line:84% If you didn't follow everything I just said, that's fine. 00:53:06.010 --> 00:53:08.800 align:middle line:90% You could start listening now. 00:53:08.800 --> 00:53:11.350 align:middle line:84% Because then I'm just going to start defining things 00:53:11.350 --> 00:53:12.730 align:middle line:90% and proving theorems about them. 00:53:12.730 --> 00:53:17.860 align:middle line:84% But you should know the path that we're 00:53:17.860 --> 00:53:22.282 align:middle line:84% on so that you don't lose the forest through the trees 00:53:22.282 --> 00:53:23.240 align:middle line:90% or something like that. 00:53:23.240 --> 00:53:24.740 align:middle line:84% I think it goes something like that. 00:53:24.740 --> 00:53:31.225 align:middle line:90% Anyways, so outer measure-- 00:53:31.225 --> 00:53:35.070 align:middle line:90% 00:53:35.070 --> 00:53:37.560 align:middle line:90% so for an interval-- 00:53:37.560 --> 00:53:40.090 align:middle line:90% 00:53:40.090 --> 00:53:42.870 align:middle line:84% so let me just note a little notation, 00:53:42.870 --> 00:53:44.130 align:middle line:90% as I used a minute ago. 00:53:44.130 --> 00:53:58.170 align:middle line:90% 00:53:58.170 --> 00:54:09.810 align:middle line:84% If I is an interval, meaning open closed, half 00:54:09.810 --> 00:54:15.790 align:middle line:84% open, infinite, little l of I denotes its length. 00:54:15.790 --> 00:54:21.360 align:middle line:84% If it's unbounded, then this just is a stand-in for infinity. 00:54:21.360 --> 00:54:27.330 align:middle line:90% 00:54:27.330 --> 00:54:30.240 align:middle line:84% And the length of the interval-- so the length 00:54:30.240 --> 00:54:37.920 align:middle line:84% of a, b both including a, b or not, or over 1 and maybe 00:54:37.920 --> 00:54:41.640 align:middle line:84% not the other, is equal to b minus a regardless 00:54:41.640 --> 00:54:44.160 align:middle line:84% if it's an open, closed, or half open interval. 00:54:44.160 --> 00:54:47.300 align:middle line:90% 00:54:47.300 --> 00:54:56.145 align:middle line:84% So for a subset of real numbers, we define its outer measure-- 00:54:56.145 --> 00:55:08.620 align:middle line:90% 00:55:08.620 --> 00:55:10.577 align:middle line:84% we define the outer measure of a. 00:55:10.577 --> 00:55:13.000 align:middle line:90% This is m star of a. 00:55:13.000 --> 00:55:23.500 align:middle line:84% This is equal to the infimum of numbers sum of lengths of I 00:55:23.500 --> 00:55:25.090 align:middle line:90% sub n. 00:55:25.090 --> 00:55:28.480 align:middle line:90% And what are the I sub n's? 00:55:28.480 --> 00:55:39.100 align:middle line:84% This is equal where I sub n is a countable collection 00:55:39.100 --> 00:55:49.790 align:middle line:84% of open intervals such that a is contained in their union. 00:55:49.790 --> 00:55:53.500 align:middle line:90% 00:55:53.500 --> 00:55:57.640 align:middle line:84% So how I compute the outer measure, 00:55:57.640 --> 00:56:03.010 align:middle line:84% so I can cover any set-- any subset of the real numbers 00:56:03.010 --> 00:56:07.100 align:middle line:90% by a union of open intervals. 00:56:07.100 --> 00:56:13.670 align:middle line:84% And so then, the outer measure is then 00:56:13.670 --> 00:56:17.360 align:middle line:84% defined as the infimum of the sum of the lengths 00:56:17.360 --> 00:56:19.010 align:middle line:90% of these open intervals. 00:56:19.010 --> 00:56:20.730 align:middle line:90% So you have your set. 00:56:20.730 --> 00:56:22.160 align:middle line:90% You cover it by open intervals. 00:56:22.160 --> 00:56:24.230 align:middle line:84% You sum up the length of the open intervals. 00:56:24.230 --> 00:56:30.100 align:middle line:84% This gives you some rough approximation to the size of a. 00:56:30.100 --> 00:56:32.540 align:middle line:84% And now if you make those intervals small, and smaller, 00:56:32.540 --> 00:56:34.290 align:middle line:84% and smaller, then you should be picking up 00:56:34.290 --> 00:56:35.640 align:middle line:90% more detailed information. 00:56:35.640 --> 00:56:39.887 align:middle line:84% And so, the infimum of some of the length 00:56:39.887 --> 00:56:41.720 align:middle line:84% of these intervals that are covering a, this 00:56:41.720 --> 00:56:43.070 align:middle line:90% gives you the outer measure. 00:56:43.070 --> 00:56:50.690 align:middle line:84% So let me give you the stupidest example ever, which is 00:56:50.690 --> 00:56:53.690 align:middle line:90% the outer measure of a point. 00:56:53.690 --> 00:56:57.590 align:middle line:84% So let's say, I don't know, the set containing just 1. 00:56:57.590 --> 00:56:58.580 align:middle line:90% So I claim that-- 00:56:58.580 --> 00:57:02.070 align:middle line:90% 00:57:02.070 --> 00:57:03.930 align:middle line:90% or the set containing 0. 00:57:03.930 --> 00:57:07.180 align:middle line:84% The outer measure of this is just 0. 00:57:07.180 --> 00:57:09.600 align:middle line:84% So the set with just one point should-- 00:57:09.600 --> 00:57:11.130 align:middle line:90% it doesn't fill anything. 00:57:11.130 --> 00:57:16.070 align:middle line:84% So it shouldn't have any measure, 00:57:16.070 --> 00:57:18.650 align:middle line:84% it shouldn't have a positive measure. 00:57:18.650 --> 00:57:25.920 align:middle line:84% At least if we're wanting to have measure 00:57:25.920 --> 00:57:27.480 align:middle line:90% be related to length. 00:57:27.480 --> 00:57:31.590 align:middle line:90% 00:57:31.590 --> 00:57:33.670 align:middle line:90% Now, why is this? 00:57:33.670 --> 00:57:39.140 align:middle line:90% Well, let epsilon be positive. 00:57:39.140 --> 00:57:41.770 align:middle line:84% What I'm going to show is that the outer measure of this set 00:57:41.770 --> 00:57:44.630 align:middle line:90% is less than epsilon. 00:57:44.630 --> 00:57:46.990 align:middle line:84% And since this is always a non-negative number, 00:57:46.990 --> 00:57:50.750 align:middle line:84% it's the infimum over a subset of positive numbers. 00:57:50.750 --> 00:57:58.090 align:middle line:84% So therefore, that infimum is always non-negative. 00:57:58.090 --> 00:58:01.390 align:middle line:84% So I should note that this is always 00:58:01.390 --> 00:58:02.980 align:middle line:90% bigger than or equal to 0. 00:58:02.980 --> 00:58:05.470 align:middle line:84% So I'm going to show that the outer measure is 00:58:05.470 --> 00:58:09.030 align:middle line:90% less than epsilon. 00:58:09.030 --> 00:58:11.550 align:middle line:84% Then the set containing only 0 is 00:58:11.550 --> 00:58:18.160 align:middle line:84% contained in the open interval epsilon over 2-- minus epsilon 00:58:18.160 --> 00:58:19.840 align:middle line:90% over 2 to epsilon over 2. 00:58:19.840 --> 00:58:25.060 align:middle line:84% And therefore, the outer measure of 0, 00:58:25.060 --> 00:58:27.640 align:middle line:84% which is the infimum of the sum of links 00:58:27.640 --> 00:58:31.150 align:middle line:84% over all collection of intervals covering a, this 00:58:31.150 --> 00:58:34.330 align:middle line:84% should be less than or equal to if I just take one of these. 00:58:34.330 --> 00:58:44.520 align:middle line:84% So length of, which equals epsilon. 00:58:44.520 --> 00:58:47.570 align:middle line:84% Now, the measure of this is always less than or equal 00:58:47.570 --> 00:58:50.040 align:middle line:84% to epsilon for all epsilon positive. 00:58:50.040 --> 00:58:51.620 align:middle line:90% So then I can take-- 00:58:51.620 --> 00:58:52.790 align:middle line:90% send epsilon to 0. 00:58:52.790 --> 00:58:59.170 align:middle line:84% And I get that this measure-- outer measure is 0. 00:58:59.170 --> 00:59:11.000 align:middle line:90% 00:59:11.000 --> 00:59:14.000 align:middle line:84% So again, the outer measure is the infimum 00:59:14.000 --> 00:59:15.770 align:middle line:84% of the sum of lengths, of intervals, 00:59:15.770 --> 00:59:19.960 align:middle line:84% of open intervals, where these open intervals cover a. 00:59:19.960 --> 00:59:25.620 align:middle line:90% 00:59:25.620 --> 00:59:28.710 align:middle line:84% Simplest one is the set containing a single point. 00:59:28.710 --> 00:59:29.940 align:middle line:90% It has measure 0. 00:59:29.940 --> 00:59:33.150 align:middle line:84% Now I bet if you sit and think for a little bit, 00:59:33.150 --> 00:59:36.000 align:middle line:84% you can then prove that the outer measure 00:59:36.000 --> 00:59:40.890 align:middle line:84% of a set containing finitely many points, this is also 0. 00:59:40.890 --> 00:59:45.140 align:middle line:90% 00:59:45.140 --> 00:59:46.640 align:middle line:84% And if you work a little bit harder, 00:59:46.640 --> 00:59:50.120 align:middle line:84% you can even prove that the measure of a countable set 00:59:50.120 --> 00:59:57.520 align:middle line:84% is also 0 just based on what we've done here, just based on-- 00:59:57.520 --> 00:59:59.967 align:middle line:90% 00:59:59.967 --> 01:00:00.550 align:middle line:90% you know what? 01:00:00.550 --> 01:00:01.510 align:middle line:90% Let's do that. 01:00:01.510 --> 01:00:02.920 align:middle line:90% That's a fun exercise. 01:00:02.920 --> 01:00:08.730 align:middle line:90% 01:00:08.730 --> 01:00:22.710 align:middle line:84% So is countable, then the measure of a equals 0. 01:00:22.710 --> 01:00:28.110 align:middle line:84% So for example, the rational numbers are countable. 01:00:28.110 --> 01:00:30.840 align:middle line:84% There's a lot of rational numbers throughout R. 01:00:30.840 --> 01:00:32.040 align:middle line:90% But they have measure 0. 01:00:32.040 --> 01:00:35.430 align:middle line:84% They don't fill anything in a certain sense. 01:00:35.430 --> 01:00:55.110 align:middle line:84% Not feel as in F-E-E-L, fill, as an F-I-L-L. 01:00:55.110 --> 01:00:57.120 align:middle line:90% So what's the proof of this? 01:00:57.120 --> 01:01:06.860 align:middle line:84% So if a is countable, then I can list the elements of a. 01:01:06.860 --> 01:01:10.370 align:middle line:84% I mean, a being countable means there's a bijection 01:01:10.370 --> 01:01:11.765 align:middle line:90% from a to the natural numbers. 01:01:11.765 --> 01:01:15.470 align:middle line:90% 01:01:15.470 --> 01:01:17.720 align:middle line:84% And I'll do the countably infinite one 01:01:17.720 --> 01:01:24.020 align:middle line:84% and leave the finite case to you. 01:01:24.020 --> 01:01:32.380 align:middle line:90% 01:01:32.380 --> 01:01:34.590 align:middle line:84% So that means there exists a bijection 01:01:34.590 --> 01:01:36.760 align:middle line:90% from a to the natural numbers. 01:01:36.760 --> 01:01:42.660 align:middle line:84% But that just means that I can list the elements of a, a1, a2, 01:01:42.660 --> 01:01:47.820 align:middle line:90% a3, a4, and so on. 01:01:47.820 --> 01:01:48.710 align:middle line:90% You can write it as-- 01:01:48.710 --> 01:01:55.110 align:middle line:90% 01:01:55.110 --> 01:01:56.680 align:middle line:90% OK. 01:01:56.680 --> 01:01:59.890 align:middle line:84% Now, I am, just like I did a minute ago, 01:01:59.890 --> 01:02:02.200 align:middle line:84% I'm going to show that the outer measure of this set 01:02:02.200 --> 01:02:04.840 align:middle line:84% is less than or equal to epsilon, where epsilon is 01:02:04.840 --> 01:02:07.220 align:middle line:90% an arbitrary positive number. 01:02:07.220 --> 01:02:11.830 align:middle line:84% And therefore, the outer measure has to be 0. 01:02:11.830 --> 01:02:13.240 align:middle line:90% Let epsilon be positive. 01:02:13.240 --> 01:02:17.820 align:middle line:90% 01:02:17.820 --> 01:02:22.565 align:middle line:84% We will show the outer measure of a, 01:02:22.565 --> 01:02:24.690 align:middle line:84% which is, again, a non-negative number is less than 01:02:24.690 --> 01:02:25.523 align:middle line:90% or equal to epsilon. 01:02:25.523 --> 01:02:28.190 align:middle line:90% 01:02:28.190 --> 01:02:30.130 align:middle line:84% And then, any number that's less than 01:02:30.130 --> 01:02:33.655 align:middle line:84% or equal to an arbitrarily small number has to be 0. 01:02:33.655 --> 01:02:39.630 align:middle line:90% 01:02:39.630 --> 01:02:52.030 align:middle line:84% For each n natural number, let I n be the interval that takes 01:02:52.030 --> 01:03:00.550 align:middle line:90% the form epsilon over 2n plus 1. 01:03:00.550 --> 01:03:01.550 align:middle line:90% I should write it this-- 01:03:01.550 --> 01:03:04.320 align:middle line:90% 01:03:04.320 --> 01:03:08.820 align:middle line:84% a sub n minus epsilon over 2n plus 1, 01:03:08.820 --> 01:03:12.810 align:middle line:84% a sub n plus epsilon over 2n plus 1. 01:03:12.810 --> 01:03:16.070 align:middle line:90% 01:03:16.070 --> 01:03:20.270 align:middle line:84% So I sub n is an open interval that 01:03:20.270 --> 01:03:26.820 align:middle line:90% contains a sub n for each n. 01:03:26.820 --> 01:03:38.910 align:middle line:84% And so, a is contained in this countable union, 01:03:38.910 --> 01:03:43.510 align:middle line:84% or this countable union of open intervals. 01:03:43.510 --> 01:03:46.150 align:middle line:84% And since the outer measure is the infimum 01:03:46.150 --> 01:03:49.630 align:middle line:84% of the sum of lengths of open intervals covering a, 01:03:49.630 --> 01:03:54.610 align:middle line:84% this implies that the measure, the outer measure of a, which 01:03:54.610 --> 01:03:59.590 align:middle line:84% again is the infimum, so it's smaller than sum of lengths 01:03:59.590 --> 01:04:01.540 align:middle line:84% of open intervals covering a, this 01:04:01.540 --> 01:04:05.290 align:middle line:84% is less than or equal to sum n equals 1 01:04:05.290 --> 01:04:12.100 align:middle line:84% to infinity of the length of I n, which equals sum from n 01:04:12.100 --> 01:04:16.450 align:middle line:84% equals 1 to infinity of now the length of each 01:04:16.450 --> 01:04:19.360 align:middle line:84% of these intervals is epsilon over 2 to the n. 01:04:19.360 --> 01:04:23.580 align:middle line:90% 01:04:23.580 --> 01:04:27.690 align:middle line:84% And the sum from n equals 1 to infinity of epsilon over 2 01:04:27.690 --> 01:04:29.770 align:middle line:90% to the n, this is just epsilon. 01:04:29.770 --> 01:04:41.170 align:middle line:84% And therefore, the outer measure is less than epsilon. 01:04:41.170 --> 01:04:51.080 align:middle line:84% Since epsilon was arbitrary and the outer measure 01:04:51.080 --> 01:04:53.930 align:middle line:84% is non-negative, that we conclude that the outer measure 01:04:53.930 --> 01:04:54.800 align:middle line:90% has to be 0. 01:04:54.800 --> 01:05:04.250 align:middle line:90% 01:05:04.250 --> 01:05:08.030 align:middle line:84% I mean, this argument, we can generalize 01:05:08.030 --> 01:05:12.950 align:middle line:84% this argument a little bit to prove something that I'm 01:05:12.950 --> 01:05:15.260 align:middle line:90% getting a little out of-- 01:05:15.260 --> 01:05:18.690 align:middle line:84% what I'm about to do is slightly out of order. 01:05:18.690 --> 01:05:21.150 align:middle line:84% But we can generalize the argument we just gave. 01:05:21.150 --> 01:05:23.010 align:middle line:84% And I want to do it now since it's there 01:05:23.010 --> 01:05:27.690 align:middle line:84% and we just did it, to prove that outer measure satisfies 01:05:27.690 --> 01:05:30.420 align:middle line:84% something that's almost like three. 01:05:30.420 --> 01:05:36.540 align:middle line:84% So and it's kind a generalization of this argument. 01:05:36.540 --> 01:05:37.920 align:middle line:84% Except for, this argument we were 01:05:37.920 --> 01:05:40.620 align:middle line:84% able to give an explicit I sub n. 01:05:40.620 --> 01:05:46.580 align:middle line:84% So we have the following theorem that, 01:05:46.580 --> 01:05:52.030 align:middle line:84% well, first off let me state that a very easy property 01:05:52.030 --> 01:05:53.920 align:middle line:84% of outer measure is that if I have 01:05:53.920 --> 01:05:58.120 align:middle line:84% two subsets of real numbers, and a is contained in b, then 01:05:58.120 --> 01:06:01.240 align:middle line:84% the outer measure of a is less than 01:06:01.240 --> 01:06:05.230 align:middle line:84% or equal to the outer measure of b. 01:06:05.230 --> 01:06:07.300 align:middle line:84% Because this just follows from the definition. 01:06:07.300 --> 01:06:12.190 align:middle line:84% Any covering of b by open intervals 01:06:12.190 --> 01:06:16.120 align:middle line:84% is a covering of a by open intervals. 01:06:16.120 --> 01:06:18.250 align:middle line:84% And therefore, the infimum has to be less than 01:06:18.250 --> 01:06:21.262 align:middle line:84% or equal to the sum of those lengths. 01:06:21.262 --> 01:06:23.720 align:middle line:84% Or I should say, then the outer measure has to be less than 01:06:23.720 --> 01:06:25.760 align:middle line:84% or equal to the sum of those links. 01:06:25.760 --> 01:06:30.050 align:middle line:84% And thus, the infimum over all of those intervals covering 01:06:30.050 --> 01:06:31.610 align:middle line:84% b, which is the outer measure, has 01:06:31.610 --> 01:06:33.970 align:middle line:84% to be bigger than or equal to the outer measure of a. 01:06:33.970 --> 01:06:37.020 align:middle line:84% So just think about it for a minute. 01:06:37.020 --> 01:06:40.640 align:middle line:84% But this should be pretty much clear from the definition. 01:06:40.640 --> 01:06:50.110 align:middle line:90% 01:06:50.110 --> 01:06:59.118 align:middle line:84% So the next theorem is that we have 01:06:59.118 --> 01:07:00.285 align:middle line:90% something that's like three. 01:07:00.285 --> 01:07:06.570 align:middle line:90% 01:07:06.570 --> 01:07:12.790 align:middle line:90% Let a and A n be a collection-- 01:07:12.790 --> 01:07:14.770 align:middle line:84% be a countable collection, I should say. 01:07:14.770 --> 01:07:31.380 align:middle line:84% So here n is a natural number, collection of subsets of R. 01:07:31.380 --> 01:07:36.150 align:middle line:84% Then the outer measure of their union-- 01:07:36.150 --> 01:07:39.960 align:middle line:84% so this is just arbitrary subsets. 01:07:39.960 --> 01:07:45.480 align:middle line:84% I mean, they don't have to be pairwise disjoint like one 01:07:45.480 --> 01:07:45.980 align:middle line:90% and three. 01:07:45.980 --> 01:07:48.140 align:middle line:84% But so, just an arbitrary collection. 01:07:48.140 --> 01:07:50.390 align:middle line:84% If I take the outer measure of the union, 01:07:50.390 --> 01:07:54.200 align:middle line:84% this is less than or equal to the sum of the outer measures. 01:07:54.200 --> 01:08:17.350 align:middle line:90% 01:08:17.350 --> 01:08:20.470 align:middle line:84% So in particular, if this collection 01:08:20.470 --> 01:08:26.760 align:middle line:84% is a collection of pairwise disjoint sets, 01:08:26.760 --> 01:08:32.640 align:middle line:84% then the measure of the hole is less than or equal-- 01:08:32.640 --> 01:08:35.850 align:middle line:84% less than or equal to the sum of the measure of the parts, 01:08:35.850 --> 01:08:41.609 align:middle line:84% which is half of the three that we want. 01:08:41.609 --> 01:08:42.899 align:middle line:90% So we're getting there. 01:08:42.899 --> 01:08:45.559 align:middle line:84% But anyways, so let's prove this. 01:08:45.559 --> 01:08:48.140 align:middle line:90% 01:08:48.140 --> 01:09:03.790 align:middle line:84% First off, if there exists an n such 01:09:03.790 --> 01:09:08.790 align:middle line:84% that it has infinite outer measure, 01:09:08.790 --> 01:09:12.930 align:middle line:84% just meaning that that subset that I'm taking the infimum over 01:09:12.930 --> 01:09:16.529 align:middle line:90% is unbounded, or-- 01:09:16.529 --> 01:09:17.149 align:middle line:90% no, no, no. 01:09:17.149 --> 01:09:19.520 align:middle line:90% That's not what I mean. 01:09:19.520 --> 01:09:29.170 align:middle line:84% But if that infimum is infinite, meaning 01:09:29.170 --> 01:09:33.850 align:middle line:84% I can't cover the set by a collection of intervals with sum 01:09:33.850 --> 01:09:47.279 align:middle line:84% of links as a finite number, or the sum is infinite, 01:09:47.279 --> 01:10:02.130 align:middle line:84% meaning it's not convergent, then this inequality is true. 01:10:02.130 --> 01:10:06.420 align:middle line:84% I mean, then I'm just saying something 01:10:06.420 --> 01:10:09.360 align:middle line:84% is less than or equal to infinity. 01:10:09.360 --> 01:10:12.427 align:middle line:84% Or now I'm treating infinity like an element 01:10:12.427 --> 01:10:13.635 align:middle line:90% of the extended real numbers. 01:10:13.635 --> 01:10:17.110 align:middle line:90% 01:10:17.110 --> 01:10:22.180 align:middle line:84% So if one of the measures is infinite, 01:10:22.180 --> 01:10:26.030 align:middle line:84% or if this series diverges and I just get an infinite number, 01:10:26.030 --> 01:10:28.310 align:middle line:84% then this inequality holds trivially. 01:10:28.310 --> 01:10:31.638 align:middle line:84% So let's just consider the case when-- 01:10:31.638 --> 01:10:34.180 align:middle line:84% so we just need to consider the case when all of the measures 01:10:34.180 --> 01:10:38.790 align:middle line:84% are finite and the sum of the measures is finite. 01:10:38.790 --> 01:10:50.930 align:middle line:84% So suppose while n be an outer measure of 01:10:50.930 --> 01:10:59.245 align:middle line:84% and the outer measure-- the sum of the outer measures converges. 01:10:59.245 --> 01:10:59.870 align:middle line:90% So it's finite. 01:10:59.870 --> 01:11:09.120 align:middle line:90% 01:11:09.120 --> 01:11:13.440 align:middle line:84% So this is an argument you have to get used to, 01:11:13.440 --> 01:11:16.440 align:middle line:84% that instead of proving the inequality that you want 01:11:16.440 --> 01:11:19.630 align:middle line:84% to prove, you prove this inequality plus epsilon, 01:11:19.630 --> 01:11:21.510 align:middle line:84% where epsilon is an arbitrary thing-- 01:11:21.510 --> 01:11:25.920 align:middle line:84% arbitrary positive number, just to give yourself a little room. 01:11:25.920 --> 01:11:28.380 align:middle line:84% And then if you're able to prove the inequality you want, 01:11:28.380 --> 01:11:31.680 align:middle line:84% plus epsilon, where epsilon is an arbitrary positive number, 01:11:31.680 --> 01:11:36.420 align:middle line:84% then the inequality holds by letting epsilon go to 0. 01:11:36.420 --> 01:11:39.320 align:middle line:90% 01:11:39.320 --> 01:11:41.990 align:middle line:84% So that's the goal is we're going 01:11:41.990 --> 01:11:45.510 align:middle line:84% to prove that inequality with an epsilon on the right-hand side. 01:11:45.510 --> 01:11:46.670 align:middle line:90% So plus an epsilon. 01:11:46.670 --> 01:11:48.230 align:middle line:90% Let epsilon be positive. 01:11:48.230 --> 01:11:53.370 align:middle line:90% 01:11:53.370 --> 01:12:01.730 align:middle line:84% For each n let collection I n sub k, 01:12:01.730 --> 01:12:15.250 align:middle line:84% kn natural number, be a collection of open intervals 01:12:15.250 --> 01:12:16.645 align:middle line:90% covering a sub n-- 01:12:16.645 --> 01:12:21.090 align:middle line:90% 01:12:21.090 --> 01:12:28.240 align:middle line:90% so I sub n sub k. 01:12:28.240 --> 01:12:32.190 align:middle line:90% 01:12:32.190 --> 01:12:39.390 align:middle line:84% And so remember, the outer measure is an infimum. 01:12:39.390 --> 01:12:42.480 align:middle line:84% So if I go above this infimum a little bit, 01:12:42.480 --> 01:12:45.840 align:middle line:84% I can always find a collection of open intervals covering 01:12:45.840 --> 01:12:48.780 align:middle line:84% the set whose sum of links is less 01:12:48.780 --> 01:12:51.480 align:middle line:84% than that infimum plus the small number. 01:12:51.480 --> 01:12:58.590 align:middle line:84% So and sum of k equals 1 to infinity of I n 01:12:58.590 --> 01:13:11.320 align:middle line:90% sub k is less than 2 to the n. 01:13:11.320 --> 01:13:15.070 align:middle line:84% Again, just because this is defined as an infimum, 01:13:15.070 --> 01:13:18.670 align:middle line:84% and so I can always for any little bit, if I go above it-- 01:13:18.670 --> 01:13:20.710 align:middle line:90% or a sub n, sorry-- 01:13:20.710 --> 01:13:22.960 align:middle line:84% if I go above it, then I should be 01:13:22.960 --> 01:13:25.360 align:middle line:84% able to find a collection of open intervals 01:13:25.360 --> 01:13:28.660 align:middle line:84% so that the sum of the links-- oh, God, what is that? 01:13:28.660 --> 01:13:33.640 align:middle line:84% Sum of links-- it's the end of the day. 01:13:33.640 --> 01:13:36.420 align:middle line:84% So I'm starting to make careless board mistakes. 01:13:36.420 --> 01:13:54.210 align:middle line:90% 01:13:54.210 --> 01:13:59.670 align:middle line:84% And now, each of these collections for each n cover 01:13:59.670 --> 01:14:03.030 align:middle line:90% the a sub n's. 01:14:03.030 --> 01:14:07.140 align:middle line:84% And therefore, the union of the a sub 01:14:07.140 --> 01:14:20.110 align:middle line:84% n's is contained in the union of n in n, k in n of I in k. 01:14:20.110 --> 01:14:25.540 align:middle line:84% So this is a collection of open intervals covering 01:14:25.540 --> 01:14:27.310 align:middle line:90% this union is a countable, yes. 01:14:27.310 --> 01:14:31.090 align:middle line:84% Because it's in one to one correspondence with n cross n. 01:14:31.090 --> 01:14:36.530 align:middle line:84% n cross n is, again, a countable set. 01:14:36.530 --> 01:14:42.620 align:middle line:84% So this is a countable collection of open intervals 01:14:42.620 --> 01:14:47.390 align:middle line:90% covering the union of a sub n. 01:14:47.390 --> 01:14:55.142 align:middle line:84% And therefore, the measure of a sub n should be less than 01:14:55.142 --> 01:14:56.600 align:middle line:84% or equal to the length of the sums. 01:14:56.600 --> 01:15:08.800 align:middle line:90% 01:15:08.800 --> 01:15:18.280 align:middle line:84% And this is equal to sum over n sum over k, I in k. 01:15:18.280 --> 01:15:23.440 align:middle line:90% And this is less than sum n-- 01:15:23.440 --> 01:15:27.220 align:middle line:84% if I'm summing in k, remember I have that-- 01:15:27.220 --> 01:15:35.700 align:middle line:84% the sum in k is less than m star of a sub n 01:15:35.700 --> 01:15:39.480 align:middle line:90% plus epsilon over 2 to the n. 01:15:39.480 --> 01:15:41.100 align:middle line:90% And now I'm summing in n. 01:15:41.100 --> 01:15:44.100 align:middle line:84% So n is going from 1 to infinity and so is k. 01:15:44.100 --> 01:15:54.260 align:middle line:84% And so this equals the sum of n, m star of a sub n plus epsilon. 01:15:54.260 --> 01:15:55.960 align:middle line:90% So I started with-- 01:15:55.960 --> 01:15:59.860 align:middle line:84% ah, I keep making silly mistakes. 01:15:59.860 --> 01:16:06.570 align:middle line:90% 01:16:06.570 --> 01:16:07.690 align:middle line:90% Sorry about that. 01:16:07.690 --> 01:16:12.240 align:middle line:84% So I started with the measure of the union of the a sub n's. 01:16:12.240 --> 01:16:14.160 align:middle line:84% It's less than or equal to the sum 01:16:14.160 --> 01:16:19.410 align:middle line:84% of the lengths of these open intervals that cover this union. 01:16:19.410 --> 01:16:22.845 align:middle line:84% Because the outer measure is the infimum over all such sums. 01:16:22.845 --> 01:16:26.190 align:middle line:90% 01:16:26.190 --> 01:16:27.680 align:middle line:90% So that's equal to this sum. 01:16:27.680 --> 01:16:29.600 align:middle line:84% How did we choose these open intervals? 01:16:29.600 --> 01:16:32.810 align:middle line:84% We chose them so that these lengths, the sum and k 01:16:32.810 --> 01:16:36.080 align:middle line:84% of these lengths gives me approximately the outer measure 01:16:36.080 --> 01:16:38.990 align:middle line:84% of each a sub n plus a little error. 01:16:38.990 --> 01:16:41.187 align:middle line:84% And I chose this error so that it's summable. 01:16:41.187 --> 01:16:42.770 align:middle line:84% If I would have just put epsilon here, 01:16:42.770 --> 01:16:48.670 align:middle line:84% I would have been summing something that can't be summed. 01:16:48.670 --> 01:16:51.130 align:middle line:90% And in the end, I get this. 01:16:51.130 --> 01:16:58.580 align:middle line:84% So I've shown that for all epsilon positive, 01:16:58.580 --> 01:17:03.800 align:middle line:84% the outer measure of the union is less than 01:17:03.800 --> 01:17:10.070 align:middle line:84% or equal to the sum of the outer measures plus epsilon. 01:17:10.070 --> 01:17:11.360 align:middle line:90% I guess, less than. 01:17:11.360 --> 01:17:14.150 align:middle line:90% 01:17:14.150 --> 01:17:16.670 align:middle line:84% And now I just let epsilon go to 0. 01:17:16.670 --> 01:17:17.990 align:middle line:90% And I conclude that-- 01:17:17.990 --> 01:17:28.750 align:middle line:90% 01:17:28.750 --> 01:17:29.725 align:middle line:90% the bound that I want. 01:17:29.725 --> 01:17:35.260 align:middle line:90% 01:17:35.260 --> 01:17:37.660 align:middle line:84% So this also gives a second proof, 01:17:37.660 --> 01:17:40.740 align:middle line:84% if you like, of what we proved a minute ago, 01:17:40.740 --> 01:17:43.650 align:middle line:84% that countable sets have measure 0. 01:17:43.650 --> 01:17:46.840 align:middle line:84% And if you-- and what we did right there, 01:17:46.840 --> 01:17:51.030 align:middle line:90% which is one element. 01:17:51.030 --> 01:17:55.170 align:middle line:84% We proved that the set containing one element 01:17:55.170 --> 01:17:57.390 align:middle line:90% has outer measure 0. 01:17:57.390 --> 01:18:02.910 align:middle line:84% Now, a countable set is equal to a union of such sets. 01:18:02.910 --> 01:18:06.090 align:middle line:84% We've shown that the outer measure of this union, 01:18:06.090 --> 01:18:08.460 align:middle line:84% which would be this countable set, is less than 01:18:08.460 --> 01:18:12.150 align:middle line:84% or equal to the sum of the outer measure of the individual points 01:18:12.150 --> 01:18:12.990 align:middle line:90% in that set. 01:18:12.990 --> 01:18:15.930 align:middle line:84% And we did, in this example, we did it for 0. 01:18:15.930 --> 01:18:18.180 align:middle line:84% But it doesn't matter that it was 0, 01:18:18.180 --> 01:18:20.490 align:middle line:84% that the outer measure of a singleton, a set 01:18:20.490 --> 01:18:21.990 align:middle line:90% with a single point, is 0. 01:18:21.990 --> 01:18:24.210 align:middle line:84% And therefore, the sum would be 0. 01:18:24.210 --> 01:18:29.290 align:middle line:84% And we would conclude that the outer measure of countable set 01:18:29.290 --> 01:18:29.860 align:middle line:90% is 0. 01:18:29.860 --> 01:18:35.710 align:middle line:84% So that's a second proof showing that the outer measure 01:18:35.710 --> 01:18:36.865 align:middle line:90% of a countable set is 0. 01:18:36.865 --> 01:18:41.990 align:middle line:90% 01:18:41.990 --> 01:18:45.290 align:middle line:84% So we've shown that outer measure is-- 01:18:45.290 --> 01:18:47.000 align:middle line:84% so this is something that's defined 01:18:47.000 --> 01:18:53.460 align:middle line:84% for every subset of real numbers and that it satisfies three. 01:18:53.460 --> 01:18:58.580 align:middle line:84% Now, I'm going to leave-- it's actually 01:18:58.580 --> 01:19:02.870 align:middle line:84% going to be an exercise in the assignment, 01:19:02.870 --> 01:19:05.930 align:middle line:84% that outer measure also satisfies four. 01:19:05.930 --> 01:19:08.390 align:middle line:84% The outer measure of a set shifted 01:19:08.390 --> 01:19:11.780 align:middle line:84% is equal to the outer measure of the original set. 01:19:11.780 --> 01:19:13.250 align:middle line:90% Why is this? 01:19:13.250 --> 01:19:15.950 align:middle line:84% A way to think about it is that if something 01:19:15.950 --> 01:19:20.000 align:middle line:84% holds for open intervals it should hold for outer measure. 01:19:20.000 --> 01:19:27.140 align:middle line:84% Because outer measure is defined in terms of sums 01:19:27.140 --> 01:19:29.000 align:middle line:90% of lengths of open intervals. 01:19:29.000 --> 01:19:32.970 align:middle line:84% So if I take an open interval and shift it, 01:19:32.970 --> 01:19:37.230 align:middle line:84% then it's length does not change. 01:19:37.230 --> 01:19:40.800 align:middle line:84% The length of ab is the same as the length of a plus 01:19:40.800 --> 01:19:42.990 align:middle line:90% x to b plus x. 01:19:42.990 --> 01:19:46.710 align:middle line:84% So it's going to be an exercise in the assignment, 01:19:46.710 --> 01:19:49.590 align:middle line:84% that outer measure, in fact, satisfies four. 01:19:49.590 --> 01:19:52.110 align:middle line:84% So what's left, and what we'll do next time, 01:19:52.110 --> 01:19:55.440 align:middle line:84% is show that if I is an interval, 01:19:55.440 --> 01:19:58.470 align:middle line:84% that the outer measure is the same 01:19:58.470 --> 01:20:01.170 align:middle line:90% as the length of the interval. 01:20:01.170 --> 01:20:04.530 align:middle line:84% And as intuitive as that is, it takes just a little bit of work 01:20:04.530 --> 01:20:06.250 align:middle line:90% to show-- 01:20:06.250 --> 01:20:09.760 align:middle line:90% not too much, but just a little. 01:20:09.760 --> 01:20:14.020 align:middle line:84% And that will complete our construction of outer measure 01:20:14.020 --> 01:20:23.160 align:middle line:84% which is this set function which satisfies one through four 01:20:23.160 --> 01:20:26.100 align:middle line:90% but not three really. 01:20:26.100 --> 01:20:29.310 align:middle line:90% It satisfies almost three. 01:20:29.310 --> 01:20:31.980 align:middle line:84% And then, once we restrict that outer measure 01:20:31.980 --> 01:20:38.610 align:middle line:84% to a certain class of subsets, certain well-behaved subsets, 01:20:38.610 --> 01:20:40.620 align:middle line:90% then we'll be able to get three. 01:20:40.620 --> 01:20:44.670 align:middle line:84% And that class of well-behaved subsets 01:20:44.670 --> 01:20:48.490 align:middle line:84% we will call Lebesgue measurable sets. 01:20:48.490 --> 01:20:48.990 align:middle line:90% All right. 01:20:48.990 --> 01:20:51.440 align:middle line:90% So I think we'll stop there. 01:20:51.440 --> 01:20:58.000 align:middle line:90%