AI Assistant
Transcript
00:03:490Paolo Guiotto: Okay, so, let's… quickly.
00:11:480Paolo Guiotto: Give a look to what we…
01:02:110Paolo Guiotto: And, introducing… Yeah, is wrong with this, and they always change this, they go…
01:13:950Paolo Guiotto: So a measurable function, and
01:18:220Paolo Guiotto: we have seen just a few examples and these characteristics. For example, this characteristic is interesting, because if we go down to the case of LeBank measure, so we have this proposition.
01:35:770Paolo Guiotto: So, let's now… The spaces are, RK…
01:43:390Paolo Guiotto: with the family of… with the Lebesgue class and the Lebesgue measure lambda K.
01:50:690Paolo Guiotto: A function F, which is continuous on RK, We are valued.
02:03:680Paolo Guiotto: is, measurable.
02:07:940Paolo Guiotto: So… Let's say that nice functions are measurable functions.
02:13:770Paolo Guiotto: And this is, just one line, because if we take,
02:19:550Paolo Guiotto: You know, last time we proved this characterization, if you take the third one.
02:25:350Paolo Guiotto: We have to check that sets where F is larger, strictly larger than A, are measurable for every A. We check exactly this. So the set where F is greater than A literally means the set of X, such that F of X is
02:43:540Paolo Guiotto: strictly larger than A. We can see this, of course, as the set of X such that f of x belongs to the interval from A to plus infinity.
02:54:240Paolo Guiotto: And this set is a set of axes, so it's a counterimage through the function f.
03:00:360Paolo Guiotto: So, is the set, usually denoted by F minus 1 of the interval from A to plus infinity, the counterimage of this.
03:10:560Paolo Guiotto: Now, there is a remarkable property, of continuous functions that says that when you, pull back an open set.
03:20:900Paolo Guiotto: The anti-image of an open set is still open.
03:24:820Paolo Guiotto: So, since F, if you have never seen it, doesn't matter, you can accept it comes with the continuity, okay? Since F is continuous.
03:35:970Paolo Guiotto: We know.
03:40:400Paolo Guiotto: That, huh?
03:42:680Paolo Guiotto: The anti-image of a set A.
03:48:940Paolo Guiotto: is open.
03:53:350Paolo Guiotto: Whenever A is, contended in, R is open.
04:00:620Paolo Guiotto: And this is the case, because this set is an open set in the real line.
04:06:370Paolo Guiotto: So, the ante image of this is open.
04:12:490Paolo Guiotto: So… F minus 1 of A plus infinity He's open.
04:20:829Paolo Guiotto: And we know that the family, the Lebec class, contains open sets, so… This is a measurable set.
04:38:420Paolo Guiotto: You see why it's sometimes convenient to have these equivalent characterizations, because maybe using this
04:47:510Paolo Guiotto: The proof for this case is a bit easier.
04:52:510Paolo Guiotto: Okay, so this says that good functions, let's say, so we have a lot of examples of measurable functions, any continuous function is a measure of fun… measurable function.
05:07:150Paolo Guiotto: Okay, so we have seen… That. So, let's see some… let's think about this. So, remark.
05:18:430Paolo Guiotto: We have seen that the indicator of any measurable set is a measurable function. So, indicator of i.
05:27:790Paolo Guiotto: is measurable.
05:30:850Paolo Guiotto: If and only if, This E is measurable.
05:36:20Paolo Guiotto: So let's, think to the case of the bag measure.
05:42:420Paolo Guiotto: Hi.
05:43:360Paolo Guiotto: So… If, X, F, mu is,
05:53:770Paolo Guiotto: Well, let's stay in dimension 1 to make things a bit easier, so R
05:58:550Paolo Guiotto: the Lebec class of R and the Lebec measure on R.
06:03:450Paolo Guiotto: indicator of, E is measurable.
06:12:100Paolo Guiotto: if and only if this set belongs to the Liberg class M1. And here, we can have a… we know that there are many, many sets here, so all open sets, closed sets.
06:26:860Paolo Guiotto: So just to show you an example, of course, there are lots of examples. If you take E equal, the set of rational numbers.
06:37:390Paolo Guiotto: This set belongs to the family M1. But now, if you have to explain why…
06:46:560Paolo Guiotto: what is the reason why this set is in the class M1? So, you know, we have a definition for the characterization of measurable sets through open sets, etc, etc. But here, we can use something much easier.
07:03:500Paolo Guiotto: Now, this set belongs into M1 because it is a measures 0 set. So let's open a parenthesis. Q belongs to M1 because
07:16:580Paolo Guiotto: The outer measure…
07:18:800Paolo Guiotto: of Q is 0. You're reminded that the class M1 contains all sets without a measure equals 0. That you may think they are irrelevant for practical purposes because they have no weight, they have no measure, but they are important because, as you will see, for example.
07:37:820Paolo Guiotto: This allows to have functions that can be extremely regular, because we can change on this kind of sets. I will detail in a second. Let's see why this is the case. And this is actually a more general fact. So let's say… general…
07:57:130Paolo Guiotto: Factor is that… Andy.
08:01:880Paolo Guiotto: Fine, it's… are countable, sect.
08:09:930Paolo Guiotto: N… pause.
08:14:110Paolo Guiotto: measure… 0, outer measure equals zero, so it belongs to the family.
08:21:550Paolo Guiotto: And this is in general for MK.
08:25:740Paolo Guiotto: So, in particular, for N1.
08:28:170Paolo Guiotto: And why this? Because if a set N is countable, and countable.
08:35:940Paolo Guiotto: Of course, find it, Is a bit…
08:40:330Paolo Guiotto: is a special case of this, no? This means that we can list points of this set with a countable index, with a naturals. So we say… we can say that this is a set of points XN with N in the set of naturals. Awesome.
08:57:460Paolo Guiotto: values XN.
09:00:340Paolo Guiotto: Now, so we can say that the outer measure of N
09:06:610Paolo Guiotto: We can see this as union of singletones, XN with N in natural.
09:15:540Paolo Guiotto: And, so this means that we can represent that measure as the measure of the countable union of these singletones.
09:25:880Paolo Guiotto: Now, of course, we assume that DXN are different, and different from N, XN is different from XM, otherwise we are…
09:35:760Paolo Guiotto: two indexes for the same element, okay? So we can always assume that they are different, so in particular, this would mean that this union is disjoint. But we know that, in principle, lambda star is not a count… is not countable additive, so
09:50:180Paolo Guiotto: We have to be a little bit more careful, but we can say that it is a countable sub-additive, so we can… because we guessed that, probably this is a measure… see if we have to prove that this is a measure zero set, we need just a bound, so this is less or equal than the sum
10:07:90Paolo Guiotto: over the n of lambda star of a singleton XN.
10:12:240Paolo Guiotto: Now, the measure of a singleton is zero.
10:16:390Paolo Guiotto: That's because you can see a singleton as the generator rectangle. So the singleton XN, which is, let's say, a point with the… so I'm doing the argument in general, so let's say XN1, XN2, etc, XNK.
10:35:340Paolo Guiotto: It will have k components, or if you want to think to the K is k equals 1, that is just… is a number. So…
10:45:310Paolo Guiotto: This set, the set made of just this point XN, is an interval, is the interval XN1, XN1, so you see.
10:56:860Paolo Guiotto: first coordinate equal just XN1, times XN2
11:03:590Paolo Guiotto: XN2, second coordinated, equal to XN2, and so on.
11:09:530Paolo Guiotto: Until the end.
11:10:850Paolo Guiotto: And the measure of this, so the outer measure, lambda star, of the singleton.
11:18:90Paolo Guiotto: Is the outer measure of this.
11:23:580Paolo Guiotto: Which is its size as a rectangle, because we know that the outer measure is already coherent with the geometrical measure, so it is the size of the same set.
11:35:580Paolo Guiotto: And this size is the product of final point minus initial pointer, so XN1 minus XN1 times XN2 minus XN2, and so on, and that's 0.
11:45:450Paolo Guiotto: Of course not. This is the point that there is no volume, no area, okay? It's just a zero stuff.
11:53:320Paolo Guiotto: So, we can say that this is also equal to 0, and therefore, at the end, we have lambda star n less or equal 0, and this means that, since it must be greater or equal than zero.
12:06:670Paolo Guiotto: The only possibility is that lambda star n is 0.
12:13:850Paolo Guiotto: Okay, so let's close the parentheses. This is to say that a countable set for the LeBague class
12:25:310Paolo Guiotto: Is, always a measurable set, and it is a measure zero set.
12:30:210Paolo Guiotto: So, these are examples. You can have any kind of countable set. We'll always have measure equal to zero. So, if you see rationales, you know, they are everywhere in the real life.
12:45:170Paolo Guiotto: They are dense in whatever is the interval you take in the real line, there are always rationals inside, and infinitely many.
12:53:200Paolo Guiotto: Nonetheless, this set has no size with respect to the measure, has measure zero. It is negligible for the measure.
13:04:350Paolo Guiotto: There are sets, but this is a bit more complicated in the real line, which are more than rational, so they are uncountable, and they still have measure zero. An example is the counter set that
13:18:160Paolo Guiotto: I asked you to try to do the exercise, maybe next time we will… we will see the solution, or in any case, I will publish the solutions of the exercises.
13:27:170Paolo Guiotto: That's a non-trivial example in dimension 1 of a set which is not countable, but still has measure 0.
13:35:920Paolo Guiotto: That said, it is in correspondence with reals, so it has the same cardinality of reals, but it is a measure zero set. In Cartesian plane, it is easier to show that there are uncountable sets with the measure 0, because if you think about if I am in the Cartesian plane, so in R2,
13:58:830Paolo Guiotto: the generate a rectangle, like this one, so there is no height. This is a measure of zero set for lambda 2, no? So lambda 2 of this is equal to 0. And this set is an interval, so it is in correspondence with the real line.
14:15:140Paolo Guiotto: And, so it's, it's uncountable. So it's much easier to find these kind of examples. In dimension 1, it's a bit tricky, and the Kanto set provides an example of that set.
14:26:510Paolo Guiotto: However, these are not essential things for us. It is just to have in mind that a measure zero set is not necessarily a single point, everyone can imagine. A single point is a measure zero set.
14:39:900Paolo Guiotto: Okay? Any finite set of points, you may expect it is a measure zero set. It is less immediate that a countable set of points is a measure zero set, and that's because of this argument.
14:52:420Paolo Guiotto: And if you want to find an uncountable sector which is negligible, that's easy. For example, if we are with the LeBerg measure in a plane, that's complicated if we are in LeBague measure, but in dimension 1.
15:08:40Paolo Guiotto: Okay, it's… it's a bit tricky. Okay, so let's now, return to discussion on functions. So, in particular.
15:18:560Paolo Guiotto: From this,
15:24:520Paolo Guiotto: it follows.
15:30:290Paolo Guiotto: That, the indicator of rationality.
15:36:30Paolo Guiotto: is Le Begre.
15:38:710Paolo Guiotto: Measurable.
15:42:130Paolo Guiotto: on R.
15:44:650Paolo Guiotto: So, since LeBank measurable functions are important, because we are going to compute integrals for these things.
15:51:860Paolo Guiotto: in a few minutes, let's say, I don't know if today, but maybe next time, we will introduce the operation of integral, and then we'll see that this function, the integral of this function as a meaning.
16:06:380Paolo Guiotto: As well as, also, the complementary. If you take a real minus rationals.
16:12:920Paolo Guiotto: Since this is the complementary of Q,
16:17:770Paolo Guiotto: So, we know a set is measurable, also its complementary is measurable, so the set of irrationals is measurable, and this indicator is measurable on R. This function is what is called the Dirichlet function.
16:32:900Paolo Guiotto: And it is usually shown in the first-year calculus causes as the example of a function which is not integral, according to Riemann sense. So it says Riemann is a good device if you have nice functions, like continuous functions.
16:51:470Paolo Guiotto: You can have some discontinuities, but not too many, okay? And this function turns out to be discontinuous at every point. There is no point where this function is continuous, because there is not the limit, no? If you fix a point, and you go to that point.
17:07:750Paolo Guiotto: The limit depends on how you go. If you go a long rationals, you get 0. If you go along irrational, you get 1. So the limit does not exist, okay? So this is the bearishly…
17:23:800Paolo Guiotto: function.
17:25:930Paolo Guiotto: that we will use, sometimes as an example.
17:33:830Paolo Guiotto: Now, the interesting thing is that if you look at this function, for example, the indicator of QR, the indicator of Q complementary, the remark is that
17:47:570Paolo Guiotto: as I say, the indicator of Q.
17:50:890Paolo Guiotto: and indicator of QC.
17:53:520Paolo Guiotto: R.
17:55:60Paolo Guiotto: Never.
17:58:180Paolo Guiotto: continuous.
18:00:420Paolo Guiotto: Okay, so these are really the opposite of the prototype of a continuous boundary. There is no point where they are continuous.
18:08:960Paolo Guiotto: Okay?
18:10:180Paolo Guiotto: Now, an interesting fact with the Lebec measure is the following. So, we have this sort of, extremes, in some sense. Continuous functions are measurable.
18:24:20Paolo Guiotto: And here we have example of extremely discontinuous functions as still measurable.
18:30:750Paolo Guiotto: Now, the interesting fact is this proposition that says that if we modify a continuous function on a measure zero set.
18:43:400Paolo Guiotto: at our pleasure, so as we like, we still get a measurable function. Of course, we won't get a continuous function, because we change the power. The value of the continuous function is unique.
18:57:990Paolo Guiotto: Wants to change a value, a single… at a single point, you surely… okay.
19:07:930Paolo Guiotto: This says that you could modify a continuous function on a measure zero set.
19:13:860Paolo Guiotto: still obtaining a measurable function. Of course, not a continuous function, okay? Well, actually, this is a little bit more general, because it does not depend on continuity. It says that if we modify a measurable function on a measure zero set.
19:30:920Paolo Guiotto: We get still immeasurable functions.
19:34:220Paolo Guiotto: Or, equivalently, if we have two functions, and they coincide, they coincide everywhere, except
19:41:890Paolo Guiotto: for a set of points where the set has measure zero, one is measurable if and only if the other is measurable. So, although these facts, they can be interpreted in the same way, you can modify a measurable function on a measure zero set, and you are still a measurable function.
20:01:880Paolo Guiotto: So this class is extremely flexible.
20:05:460Paolo Guiotto: You're modifying a single point, a continuous function, you don't have the continuous function anymore, you see the difference?
20:11:820Paolo Guiotto: So the concept of continuous function is extremely rigid. You change one single value, you lose continuity. Here, you could change a lot a function, you don't lose measurability. So this means that this property is a very weak property, okay?
20:30:100Paolo Guiotto: It… it… it does not suffer.
20:32:840Paolo Guiotto: Particularly modifications of, of, of, functions. So let's say in this format, So, let…
20:43:280Paolo Guiotto: This, be careful, because as you will see, this is a property I will start proving for the bag measure.
20:51:730Paolo Guiotto: It is not just for the bag measure, but there is something which for the bag measure is true, and in general, you cannot be sure it is true. So there is a key property behind this. So let's state for the bag measure.
21:05:600Paolo Guiotto: So let's have, NG be defined some domain E, Real value, though.
21:13:660Paolo Guiotto: Be such that,
21:18:30Paolo Guiotto: So, that E is contained in our K.
21:21:470Paolo Guiotto: be such that.
21:27:880Paolo Guiotto: How to say, the set…
21:33:370Paolo Guiotto: of points where the two are different, so the set of X.
21:38:110Paolo Guiotto: in E, such that F of X… is different, from G of X,
21:45:80Paolo Guiotto: So this is the set of points where the two functions are different. Let's call it N is S.
22:12:110Paolo Guiotto: Steven.
22:19:630Paolo Guiotto: the sectors.
22:43:60Paolo Guiotto: So…
23:03:850Paolo Guiotto: True.
23:09:70Paolo Guiotto: Lebec management.
23:13:600Paolo Guiotto: equals zero. So, lambda… I'm the king.
23:29:150Paolo Guiotto: of N equals 0.
23:33:170Paolo Guiotto: Then the property says that, then… F is measurable.