Title: Sigma Algebras
Series: Measure Theory
YouTube-Title: Measure Theory 1 | Sigma Algebras
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1 00:00:00,120 –> 00:00:06,390 Welcome to measure theory! This is part 1 of a series where I want to give you an introduction
2 00:00:06,390 –> 00:00:13,890 into measures and integrals and where we will prove some interesting results, in the end, which
3 00:00:13,890 –> 00:00:20,520 are used very often in mathematical topics. The motivation is indeed the famous Lebesgue
4 00:00:20,520 –> 00:00:28,140 integral. And to define the Lebesgue integral, we need a meaningful notion of a measure. So let’s
5 00:00:28,140 –> 00:00:36,720 start here with the real line. Now we can look at subsets on the real line and ask how to measure
6 00:00:36,720 –> 00:00:45,990 this subset. Or in other words what is the measure of this subset? And this is what Measure Theory is
7 00:00:45,990 –> 00:00:52,860 about: we want to give the subsets a meaningful measure or, in other words, a generalized volume.
8 00:00:52,860 –> 00:01:00,360 Or, in this case of the real line, a generalized length. The notion of length is what you know
9 00:01:00,360 –> 00:01:07,290 from the real line if you look at intervals. If you have an interval in the real line from a to b,
10 00:01:07,290 –> 00:01:15,750 (you would write it as a to b, and the whole subset is this), then you would say this has a length of
11 00:01:15,750 –> 00:01:23,670 b minus a. The natural question is now: what do we do if the subset looks more complicated than such an
12 00:01:23,670 –> 00:01:29,940 easy interval? How can we then calculate the length? There immediately, the Measure Theory comes in.
13 00:01:29,940 –> 00:01:36,960 And maybe we also want to deal with different notion of lengths. So we want to generalize this
14 00:01:36,960 –> 00:01:43,440 natural length. So maybe depending on the problem, we want to choose different lengths or different
15 00:01:43,440 –> 00:01:49,500 definitions of the length, so giving some weights at different parts to make our problem easier.
16 00:01:49,500 –> 00:01:55,680 Here you immediately see, the real line gives us a lot of motivation why you want to
17 00:01:55,680 –> 00:02:01,470 measure subsets of the real line. But obviously, we don’t want to stop there, we also want to measure
18 00:02:01,470 –> 00:02:09,360 areas, so in R^2, or even higher dimensional volumes, for example, normal volumes in three dimensions, and
19 00:02:09,360 –> 00:02:17,560 so on. So therefore it makes sense to immediately start an abstract measure theory, which now means we
20 00:02:17,560 –> 00:02:26,860 just look at an abstract set, and we call it just X. So X is just set. And for the set, we want to
21 00:02:26,860 –> 00:02:34,090 measure the volume, the generalized volume, of the subsets. So we will define a map which we will
22 00:02:34,090 –> 00:02:40,270 call later just a “measure”. Of course, this map should fulfill some rules but I will talk about
23 00:02:40,270 –> 00:02:48,220 this later. First let us start on the set level. Because we want to measure the subsets of X, it’s
24 00:02:48,220 –> 00:02:56,200 good to start with the power set of X. The power set is just the set of all subsets of X.
25 00:02:56,200 –> 00:03:03,460 And to give you a short reminder, let’s do an example. If we have the set with two elements (so let’s call
26 00:03:03,460 –> 00:03:12,220 the elements just lowercase a and b), then we can write down the power set. The empty set is always
27 00:03:12,220 –> 00:03:19,180 a subset of the set so, we always have the empty set in the power set. And the set itself is also
28 00:03:19,180 –> 00:03:27,310 a trivial subset. So X is in the power set. Now we have only two elements in the set. So we can’t form so
29 00:03:27,310 –> 00:03:35,320 many subsets, so the only possible way would be: okay choose one element, so a, and form a subset,
30 00:03:35,320 –> 00:03:43,180 and choose the other one. And in fact, these are all the subsets, so we now have defined the whole
31 00:03:43,180 –> 00:03:50,920 power set. So this is what you have to know. You have to know how to form subsets of a given set.
32 00:03:50,920 –> 00:03:59,740 And now we can give the definition for measurable sets. When I say “measurable set”, you may recognize
33 00:03:59,740 –> 00:04:07,930 immediately the idea that, maybe, we don’t have to measure all the subsets we can form but only some.
34 00:04:07,930 –> 00:04:15,040 If we have quite an amount of subsets we can and want to measure, maybe that’s good enough for our
35 00:04:15,040 –> 00:04:22,660 theory. And therefore we call such sets “measurable”. We will later see that we indeed have to do that
36 00:04:22,660 –> 00:04:31,030 because if we want to generalize even this easy measure, given by the length, to all the subsets,
37 00:04:31,030 –> 00:04:39,700 it’s not possible. Yeah, it’s only possible if we choose good sets as subsets. For these sets,
38 00:04:39,700 –> 00:04:49,480 we can generalize this length in a meaningful way. In the sense, we now look at a subset of the power
39 00:04:49,480 –> 00:04:56,980 set. So we look at a family of subsets that could be the whole power set (so there could be equality here)
40 00:04:56,980 –> 00:05:06,040 but, in general, we would have a smaller family of such subsets of X. So keep in mind
41 00:05:06,040 –> 00:05:14,710 this fancy A has as elements subsets of X. And such a collection is called a “Sigma algebra”
42 00:05:14,710 –> 00:05:22,840 if it fulfills the following rules. And these will be three rules. Please keep in mind that
43 00:05:22,840 –> 00:05:28,540 this notion of a Sigma algebra is indeed the important definition in the whole measure theory.
44 00:05:28,540 –> 00:05:35,830 and therefore we start with this here. The elements in the Sigma algebra will be called the “measurable sets”.
45 00:05:35,830 –> 00:05:43,060 So these are the sets we can measure in the end. Therefore, we immediately have the first rule
46 00:05:43,060 –> 00:05:50,620 here because we want to measure the easiest sets, which will be the empty set (also here in the
47 00:05:50,620 –> 00:05:58,450 power set given) and the whole set itself. So we want these two sets as elements in our Sigma algebra
48 00:05:58,450 –> 00:06:06,220 because we want these sets to be measurable. In the same sense, we get out the next rule:
49 00:06:06,220 –> 00:06:14,470 So what happens if we know we can measure a set A? So A in this curly A. Then we want to be able
50 00:06:14,470 –> 00:06:25,090 to also measure the complement of the set. And I denote the complement by A^c, so A^c means X without A itself.
51 00:06:25,090 –> 00:06:33,580 So this should also be a measurable set, so in the Sigma algebra A. To visualize this maybe
52 00:06:33,580 –> 00:06:46,040 a short sketch. So we have here an arbitrary set X and inside also some subset A. Measurable now means
53 00:06:46,040 –> 00:06:54,650 we know we can give a sensible generalized volume to the set A. If we know this generalized volume, we
54 00:06:54,650 –> 00:07:02,540 should also know the generalized volume outside. But this means, the complement so A^c should also
55 00:07:02,540 –> 00:07:11,810 be a measurable set. Therefore this rule (b) totally makes sense. This is what we naturally need.
56 00:07:11,810 –> 00:07:17,900 In the same sense, the third tool comes in. One could say that this third rule comes in from a
57 00:07:17,900 –> 00:07:26,570 measure process point of view. And it also gives the “Sigma” in the Sigma algebra a meaning. However, maybe,
58 00:07:26,570 –> 00:07:34,040 I first tell you what the rule says and then we can discuss where it comes from. So we start here with
59 00:07:34,040 –> 00:07:40,760 countably many subsets, which means we have A_i where the index i goes through all natural numbers.
60 00:07:40,760 –> 00:07:47,180 We could repeat the same set here, so then we only have finitely many subsets chosen,
61 00:07:47,180 –> 00:07:54,500 but the important thing is, if we have infinitely many, they are countable. Then we can look at the
62 00:07:54,500 –> 00:08:01,550 union of all the sets. So I can write the union symbol going from 1 to infinity. This defines us
63 00:08:01,550 –> 00:08:09,800 again a subset of X. And the claim is now: this is also in our Sigma algebra. This means that we
64 00:08:09,800 –> 00:08:17,300 can’t leave the Sigma algebra by using the normal union, so union of two sets and even not if we use
65 00:08:17,300 –> 00:08:26,360 a countable union of infinitely many sets. Maybe we visualize that again in a short picture. This
66 00:08:26,360 –> 00:08:38,120 is again our set X and we have a subset A inside. Now assume we have measurable sets inside (so given
67 00:08:38,120 –> 00:08:48,950 as these squares or rectangles, so this would be A_1 and then we have here A_2 and so on). The idea
68 00:08:48,950 –> 00:08:58,650 would be now that we can form the set A out of a countable union of the smaller sets A_i. If the
69 00:08:58,650 –> 00:09:05,400 blue sets are measurable, which means they have generalized volume, then the generalized volume
70 00:09:05,400 –> 00:09:12,960 of A should be the limit of the sum of all these generalized volumes. Or speaking of areas: if you
71 00:09:12,960 –> 00:09:20,850 add up all the areas inside we can form, then you should get out the area of A. And in order to speak
72 00:09:20,850 –> 00:09:29,160 of an area or a generalized volume of A, we need that the set A is measurable. So it should be an element in
73 00:09:29,160 –> 00:09:35,160 a Sigma algebra. So this countable union should also be an element in a Sigma algebra. And these
74 00:09:35,160 –> 00:09:42,630 are all the rules. All systems of subsets with these three rules now are called Sigma algebras.
75 00:09:42,630 –> 00:09:50,400 To close the definition I now write down what I told you the whole time: An element in
76 00:09:50,400 –> 00:09:57,870 the Sigma algebra is called a measurable set. So “Sigma algebra” and “measurable” are important
77 00:09:57,870 –> 00:10:04,380 notions in this Measure Theory, here. “Measurable” is given with respect to a given Sigma algebra,
78 00:10:04,380 –> 00:10:13,290 therefore sometimes also A is in the definition of “measurable” so it’s called “A-measurable” if we
79 00:10:13,290 –> 00:10:22,650 should emphasize which Sigma algebra is used here. So now, of course, we need some examples.
80 00:10:22,650 –> 00:10:30,300 We know that a Sigma algebra needs at least two elements, namely the empty set and set X itself.
81 00:10:30,300 –> 00:10:37,410 And this is always smallest possible Sigma algebra. So A defined with these two elements
82 00:10:37,410 –> 00:10:45,510 is a Sigma algebra because (b) and (c) in the rules are trivially fulfilled. The complements are in
83 00:10:45,510 –> 00:10:53,430 and also all possible unions you can form with these two elements are also in. Hence, this is the
84 00:10:53,430 –> 00:11:00,210 smallest possible Sigma algebra. The question is now what is the largest one but this is also easy
85 00:11:00,210 –> 00:11:08,490 to see because the power set itself fulfills, also trivially, all these rules. Because by definition,
86 00:11:08,490 –> 00:11:14,190 all possible subsets are immediately in the powerset. So you can’t leave it with the complements,
87 00:11:14,190 –> 00:11:20,430 and also not with the unions. No matter if they countable and uncountable. Therefore this would
88 00:11:20,430 –> 00:11:27,900 be the best case scenario that we can measure all possible subsets. However I already told you for
89 00:11:27,900 –> 00:11:34,530 important examples we can’t fulfill this. And therefore our sigma-algebra will lie between
90 00:11:34,530 –> 00:11:42,150 these two extrema. Of course, it would be nice to have a lot of measurable sets therefore the
91 00:11:42,150 –> 00:11:49,110 rule would be to get as close as possible to the second case, to the power set. But this is what we
92 00:11:49,110 –> 00:11:55,290 will do in later videos here in this series. So maybe that’s good enough for the introduction here.
93 00:11:55,290 –> 00:12:01,530 Now you know what a sigma algebra is. And next time we will talk more about measures and
94 00:12:01,530 –> 00:12:08,130 define what a measure is. Therefore thank you very much for listening and see you next time.