00:00 Introduction

00:58 Measuring lengths

03:00 Example power set

03:51 Definition sigma-algebra

10:20 Example for sigma-algebras

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.

Q1: For the set $X = { 2 } $, we can define the power set $\mathcal{P}(X)$. Which answer is correct?

A1: $\mathcal{P}(X) = { { 2 } }$

A2: $\mathcal{P}(X) = { { \emptyset, 2 } }$

A3: $\mathcal{P}(X) = { \emptyset, { 2 } }$

Q2: For the set $X = { 5, 9 }$, the collection $ \mathcal{A} = \big{ \emptyset, X, {5 }, { 9 } \big}$ forms a $\sigma$-algebra.

A1: True

A2: False

Q3: Which of these properties is not in the definition of a $\sigma$-algebra $\mathcal{A} \subseteq \mathcal{P}(X)$

A1: $\emptyset, X \in \mathcal{A} $

A2: $A\in \mathcal{A} ~ \Longrightarrow ~ A^c \in \mathcal{A} $

A3: $A_i \in \mathcal{A}$ for all $ i \in \mathbb{R}$ $ ~ \Longrightarrow ~ \bigcup_{i \in \mathbb{R}} A_i \in \mathcal{A} $

Q4: Is power set of $X$ always a $\sigma$-algebra?

A1: No, there are counterexamples.

A2: Yes, it is always the smallest possible one.

A3: Yes, it is always the largest possible one.