Information about Measure Theory - Part 3

  • Title: What is a measure?

  • Series: Measure Theory

  • YouTube-Title: Measure Theory 3 | What is a measure?

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    1 00:00:00,060 –> 00:00:08,220 Hello and welcome back to measure theory. This  is part 3 and today we will finally start talking  

    2 00:00:08,220 –> 00:00:14,340 about what measure is. I will start talking  about the definition and then I explain you  

    3 00:00:14,340 –> 00:00:21,870 all the details. Afterwards then I can show you  some examples for measures. To summarize in this  

    4 00:00:21,870 –> 00:00:28,590 video the question would be: what is a measure?  To answer this question let’s immediately start  

    5 00:00:28,590 –> 00:00:36,750 with the definition. What we need is a set X and  a sigma-algebra on the set X and let’s call this  

    6 00:00:36,750 –> 00:00:44,220 Sigma Algebra just A. And such a pair is then just  called a measurable space. So nothing special,  

    7 00:00:44,220 –> 00:00:51,750 it just means you have a set X and you fixed one  Sigma algebra on this set. And by now you know  

    8 00:00:51,750 –> 00:00:59,910 that a sigma algebra is a special collection of  subsets of this set X. Now we will look at special  

    9 00:00:59,910 –> 00:01:09,060 maps that are defined on the Sigma algebra A. And  such map I will always call by our lowercase mu.  

    10 00:01:09,060 –> 00:01:17,580 So it’s defined on the Sigma A and maps into  the positive real numbers. However we include  

    11 00:01:17,580 –> 00:01:26,040 some small detail: on the one side we include 0,  so 0 is allowed, and then we go to infinity, so  

    12 00:01:26,040 –> 00:01:33,060 this would be our normal interval: so we just look  at a positive real line including 0. But here in  

    13 00:01:33,060 –> 00:01:39,630 the measure theory, we will change this a little  bit we also include infinity. OK, this might look  

    14 00:01:39,630 –> 00:01:46,710 a little bit strange, and it is, because we just  include one new symbol, so this means, we have our  

    15 00:01:46,710 –> 00:01:54,210 positive real line where we include the 0, so this  is normal, but we also include a new symbol and we  

    16 00:01:54,210 –> 00:02:01,230 just call this symbol infinity. And to shorten  this notation, we just write here also infinity  

    17 00:02:01,230 –> 00:02:08,580 is included in the interval. This means that it is  just a short notation for saying that we have also  

    18 00:02:08,580 –> 00:02:16,450 a symbol infinity involved. How to calculate with  this new symbol, I tell you later. First I want to  

    19 00:02:16,450 –> 00:02:23,680 tell you how such a map is now called. Maybe not  so surprising such a map is now called a measure.  

    20 00:02:23,680 –> 00:02:31,750 However only if the following two conditions are  fulfilled. To see what these two rules should be,  

    21 00:02:31,750 –> 00:02:40,570 maybe recall what we want. We want to measure  subsets of this set X, which means we want to  

    22 00:02:40,570 –> 00:02:48,820 give a volume to such a subset. So a generalized  length or generalized volume. Therefore makes  

    23 00:02:48,820 –> 00:02:54,670 totally sense to restrict ourselves to the  positive real numbers where we also include zero  

    24 00:02:54,670 –> 00:03:01,660 and the symbol infinity; which means we could also  have a volume of zero and maybe also an infinite  

    25 00:03:01,660 –> 00:03:09,460 volume. But all other volumes should be positive  numbers. From this, we immediately get our first  

    26 00:03:09,460 –> 00:03:17,440 property because we know that the empty set is a  subset of X and in all Sigma algebras A involved.  

    27 00:03:17,440 –> 00:03:24,340 Therefore we want to measure this empty set or we  want to give it a volume. But the only sensible  

    28 00:03:24,340 –> 00:03:33,190 volume we can give the empty set would be zero.  If there are no elements involved, the generalized  

    29 00:03:33,190 –> 00:03:39,760 volume should be zero. Okay for the start not so  complicated so let’s go to our second property,  

    30 00:03:39,760 –> 00:03:48,520 this property should fix the idea that we can  add up volumes. Or in other words, if we have  

    31 00:03:48,520 –> 00:03:58,750 a subset given (so maybe just in this rectangle),  then we could split it up into subsets (so maybe  

    32 00:03:58,750 –> 00:04:10,060 in this way maybe here line and here line). So I  would call this A1, this is A2, this A3, A4, and  

    33 00:04:10,060 –> 00:04:19,600 A5. Now if we add up all these separate volumes,  we should end at our original volume. And this  

    34 00:04:19,600 –> 00:04:26,470 is the condition I can write down. So adding up  all the volumes where we start with i equals to 1  

    35 00:04:26,470 –> 00:04:33,910 and we’ll end with five, or we just write an N in  this generality. So we’re adding up the volumes,  

    36 00:04:33,910 –> 00:04:43,000 which means I add up mu of A_i. And we also  know, we chose the sets to be disjoint, pairwise  

    37 00:04:43,000 –> 00:04:50,890 disjoint, because we want a decomposition of our  original set. Pairwise disjoint means now that  

    38 00:04:50,890 –> 00:05:00,760 A_i intersected with A_j is the empty set if the  indices do not coincide. So i is not equal to j.  

    39 00:05:00,760 –> 00:05:07,660 Now you see something’s missing in our condition  but we can fix that immediately because we know  

    40 00:05:07,660 –> 00:05:16,630 this should be the original volume. So I can write  this as the volume of the union of all these sets.  

    41 00:05:16,630 –> 00:05:26,950 So I write here i equals to one going to N and  here I have my A_i. This is the union that gives  

    42 00:05:26,950 –> 00:05:34,480 us indeed our original green set here. And this  condition should be satisfied no matter which  

    43 00:05:34,480 –> 00:05:43,600 sets we chose. So I just write this should hold  for all A_i out of our Sigma algebra. And this  

    44 00:05:43,600 –> 00:05:51,100 property is what one calls just additive. It tells  you that if you have a finite union, you can split  

    45 00:05:51,100 –> 00:05:59,200 that up into a finite sum of the volumes. However  I should tell you, this is not the full story. We  

    46 00:05:59,200 –> 00:06:06,190 also want to include the intuition that we can  also approximate volumes. Maybe I explain it  

    47 00:06:06,190 –> 00:06:11,830 again in the picture. So if you want to calculate  the volume of this rectangle, we can split it up  

    48 00:06:11,830 –> 00:06:23,080 again into subsets so this is my set A_1. Then I  also choose an A_2 here and A_3 here and then I go  

    49 00:06:23,080 –> 00:06:32,350 on and on. So this would be my A_4 and A_5 and  you see I get smaller and smaller. But you see  

    50 00:06:32,350 –> 00:06:41,320 I get again the decomposition of my original set  into disjoint subsets. However, I have infinitely  

    51 00:06:41,320 –> 00:06:49,150 many subsets now. But they are countable, which  means I have a sequence of subsets. Now you know,  

    52 00:06:49,150 –> 00:06:57,970 I can still form the union of all the subsets to  get out my original set. Which means I put in here  

    53 00:06:57,970 –> 00:07:06,910 an infinity symbol to denote this union. And if  I add up the infinitely many unions, I should  

    54 00:07:06,910 –> 00:07:14,560 also get out the original volume. So instead of  a finite sum, I have now a series here; but it’s  

    55 00:07:14,560 –> 00:07:22,180 the series of non-negative numbers. Therefore to  denote this countable infinite additive rule, we  

    56 00:07:22,180 –> 00:07:30,070 usually call it sigma additive. And please recall  we had the same idea in the definition of the  

    57 00:07:30,070 –> 00:07:37,180 Sigma algebra: we want this union, this countable  infinity union, also be in the Sigma algebra. And  

    58 00:07:37,180 –> 00:07:43,240 therefore this is also well-defined because if we  choose elements from the Sigma algebra, we know  

    59 00:07:43,240 –> 00:07:49,360 that this countable union is also in a sigma  algebra. Therefore we can calculate mu of this  

    60 00:07:49,360 –> 00:07:56,620 one. And now we have it this is the definition  of a measure. Maybe let’s summarize that:  

    61 00:07:56,620 –> 00:08:04,030 a measure has to live on a Sigma algebra. It does  not have to be the whole power set (it could be),  

    62 00:08:04,030 –> 00:08:09,940 but in general we will see we can’t do it for  the whole power set. This just means that we  

    63 00:08:09,940 –> 00:08:17,230 want to measure some meaningful subsets from our  set X. Measuring subsets now means giving them  

    64 00:08:17,230 –> 00:08:23,890 a generalized volume, so it makes sense to give  them a volume in the non-negative numbers where  

    65 00:08:23,890 –> 00:08:30,700 we also include a symbol infinity. And then the  two properties generalize the ideas from a volume  

    66 00:08:30,700 –> 00:08:38,080 measure. The first thing tells you nothing should  have a zero volume and the second tells you that  

    67 00:08:38,080 –> 00:08:45,820 you can calculate the volume by splitting it up  into countable many subsets. Now let’s assume you  

    68 00:08:45,820 –> 00:08:53,430 have a measure chosen, then we can also fix that  in our informations here. You would write X with  

    69 00:08:53,430 –> 00:09:00,690 the Sigma algebra A and with our measure on the  Sigma algebra A: mu. And this is what we call a  

    70 00:09:00,690 –> 00:09:07,920 measure space. This is of course a very important  notion because the measure space is the space we  

    71 00:09:07,920 –> 00:09:15,150 work in. Very good. Now you learned what such  a general measure space is. And now we can talk  

    72 00:09:15,150 –> 00:09:24,570 about some simple examples. Let us fix for all  the examples, an arbitrary set X and also a Sigma  

    73 00:09:24,570 –> 00:09:30,120 algebra on the set X. And maybe we start with the  best case scenario that we can choose the whole  

    74 00:09:30,120 –> 00:09:38,760 power set for the Sigma algebra. The first measure  is very important and very easy, it’s called the  

    75 00:09:38,760 –> 00:09:45,450 counting measure. No matter what set X is, you  can always define this counting measure. Simply  

    76 00:09:45,450 –> 00:09:53,130 by setting that the measure of an arbitrary subset  a is defined in the following way and I consider  

    77 00:09:53,130 –> 00:10:01,050 two cases here. The first case would be that A  has only finitely many elements. In this case,  

    78 00:10:01,050 –> 00:10:07,410 I want the measure to be this number, so I can  just write down the number of elements just by the  

    79 00:10:07,410 –> 00:10:14,250 number symbol. So this is a well-defined natural  number or zero. So we don’t have a problem,  

    80 00:10:14,250 –> 00:10:21,510 defining exactly this. Okay for the other case,  I set it to infinity. Which means if A has not  

    81 00:10:21,510 –> 00:10:28,830 finitely many elements, I say it has infinitely  many elements. There you see: it makes sense to  

    82 00:10:28,830 –> 00:10:35,520 use our symbol infinity. What you can now show is  that this defines really a measure, so it fulfills  

    83 00:10:35,520 –> 00:10:41,490 these two rules. So that the empty set gets sent  to zero is immediately clear because the empty  

    84 00:10:41,490 –> 00:10:48,480 set has finitely many elements and the number of  elements is zero. So no problem there and also  

    85 00:10:48,480 –> 00:10:55,470 the Sigma additivity, you can easily show if you  just deal with finite sets. If you deal with the  

    86 00:10:55,470 –> 00:11:01,140 infinite sets, it’s also easy to show but then you  need to know what are the calculations rules when  

    87 00:11:01,140 –> 00:11:08,890 I deal with this infinity symbol. And this is  what I now want to show you. So just the basic  

    88 00:11:08,890 –> 00:11:14,980 calculation rules. The idea is of course thinking  in the volumes. So if you have one volume x,  

    89 00:11:14,980 –> 00:11:22,810 and then you add up an infinite volume (so you add  up infinity), you also should get out to infinity  

    90 00:11:22,810 –> 00:11:30,640 again. And this should hold for all x in our set.  So also for the symbol infinity. Or other words  

    91 00:11:30,640 –> 00:11:36,910 infinity plus infinity is always infinity. In the  same way we can do this for the multiplication:  

    92 00:11:36,910 –> 00:11:46,270 so x times infinity should be also defined as  infinity. However now be careful, I want to  

    93 00:11:46,270 –> 00:11:54,130 exclude 0 now. So if I multiply a positive number  with infinity, we get out to infinity again, but  

    94 00:11:54,130 –> 00:12:01,840 not for 0. For the special case 0 times infinity,  there are different conventions. In general, you  

    95 00:12:01,840 –> 00:12:08,140 would just say this is undefined because it could  mean anything. However in most cases in measure  

    96 00:12:08,140 –> 00:12:14,380 theory, it’s also nice to have a definition  for this combination of the symbols, and we  

    97 00:12:14,380 –> 00:12:22,000 set it to 0. However keep in mind, this is not  always applicable and, outside of measure theory,  

    98 00:12:22,000 –> 00:12:29,620 it could be completely wrong. Often this occurs  if we want to multiply two volumes. Then let’s  

    99 00:12:29,620 –> 00:12:37,030 go to the next example and this one is called the  Dirac measure. Maybe for this one visualize your  

    100 00:12:37,030 –> 00:12:47,560 set X here where we choose one fixed point, so we  choose here maybe a point p inside X. And now we  

    101 00:12:47,560 –> 00:12:54,010 just want that the whole measure is concentrated  in only this point. The usual notation one chooses  

    102 00:12:54,010 –> 00:13:01,540 for this measure is a delta where we have an index  p to denote the point. Now for a given subset A,  

    103 00:13:01,540 –> 00:13:11,770 we also define it with two cases. It is either 1  or 0. The idea we could also sketch in our drawing  

    104 00:13:11,770 –> 00:13:19,810 here. So if this is our set A, we see that p is  inside the set A. And if we want to volume to be  

    105 00:13:19,810 –> 00:13:26,680 concentrate at the point p, we now would say: okay  this set has measure 1. So it doesn’t matter how  

    106 00:13:26,680 –> 00:13:34,630 small the set is. As long as the special point  p is inside, it has measure 1. But if P is not  

    107 00:13:34,630 –> 00:13:41,320 inside the set, then it has measure 0. A good  visualization would be to think of this point as  

    108 00:13:41,320 –> 00:13:48,190 a point charge. The whole charge is in the point  but if you look at the surrounding you would give  

    109 00:13:48,190 –> 00:13:55,900 also the surrounding exactly this charge. Okay so  these were two measures that work on every set X,  

    110 00:13:55,900 –> 00:14:05,050 so in particular also for our special case R^n.  Or in other words these measures don’t measure  

    111 00:14:05,050 –> 00:14:13,660 the normal volume you, for example, have in R^3.  They are indeed generalized measures. But we know  

    112 00:14:13,660 –> 00:14:20,770 we also want to have this normal volume measure in  R^n. Therefore the exercise for measure theory is  

    113 00:14:20,770 –> 00:14:30,280 in particular to find a measure on X equal to R^n  that has some nice properties. The first property  

    114 00:14:30,280 –> 00:14:37,570 would fix that it measures the normal volume.  This means that if I put in the unit cube in  

    115 00:14:37,570 –> 00:14:45,040 the measure, which is a cube that has length 1 in  all directions, then I want to get out the volume  

    116 00:14:45,040 –> 00:14:52,660 1 as well. And the second property means it does  not matter where we measure the volume in space.  

    117 00:14:52,660 –> 00:15:01,510 In other words, the measure is invariant under  translations. So I can write that as x plus our  

    118 00:15:01,510 –> 00:15:10,780 set A is equal to the volume of the set A. And  this holds for all translation vectors x in R^n.  

    119 00:15:10,780 –> 00:15:19,690 Also visualize always this property in a short  picture. If you have your volume here so maybe  

    120 00:15:19,690 –> 00:15:27,460 this might be the set A. And now I translate all  the points with a fixed vector x. Then I find the  

    121 00:15:27,460 –> 00:15:36,600 new set x plus A here. So this is my set x plus A.  And of course this should have the same volume in  

    122 00:15:36,600 –> 00:15:43,890 our surrounding space. Of course this is not true  for an abstract generalized measure from before.  

    123 00:15:43,890 –> 00:15:49,800 But it should be true for our measure that we  want to find that generalizes the normal volume  

    124 00:15:49,800 –> 00:15:57,990 measure in R^n. However keep in mind: we only know  how to measure such cubes or cuboids and not how  

    125 00:15:57,990 –> 00:16:04,230 to measure such an arbitrary subset. And that’s  the idea that we want to extend this notion and  

    126 00:16:04,230 –> 00:16:11,670 still conserve these two properties. Later we will  see that we can indeed define a measure with this  

    127 00:16:11,670 –> 00:16:17,550 two properties and that is what we then call the  Lebesgue measure. And in the next video you will  

    128 00:16:17,550 –> 00:16:25,020 see that we can’t choose the whole power set as  a sigma-algebra. We have to choose a smaller one  

    129 00:16:25,020 –> 00:16:32,070 such that we can conserve these two properties.  And there you will see that we can easily work in  

    130 00:16:32,070 –> 00:16:38,940 the Borel sigma-algebra. Well very good. That’s  all I wanted to tell you today and I hope that  

    131 00:16:38,940 –> 00:16:45,750 helped you a little bit. For the next videos,  the real measure theory can start because now we  

    132 00:16:45,750 –> 00:16:53,220 have the notion what a measure and a measure space  is. So thank you very much and see you next time!

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