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00:02:110Annalisa Cesaroni: Hello, Laura!
00:07:850Annalisa Cesaroni: That's cool.
00:18:855Annalisa Cesaroni: No.
00:25:62Annalisa Cesaroni: okay. Hello. Last time we saw
00:30:710Annalisa Cesaroni: the definition of
00:33:740Annalisa Cesaroni: measure, theoretic boundary was set, you know.
00:37:270Annalisa Cesaroni: And actually it was
00:41:170Annalisa Cesaroni: So I
00:43:710Annalisa Cesaroni: measure theoretic part boundary of a set.
00:47:820Annalisa Cesaroni: And actually.
00:50:830Annalisa Cesaroni: I
00:52:900Annalisa Cesaroni: I defined the notion of measure theoretic boundary starting by defining the notion of density of a set. So I fixed a set, and I defined a measure associated to a set which is a Borel measure defined as the Lebeck measure. O for every Borel set a. I define this measure as the Lebeck measure of a intersected E,
01:20:340Annalisa Cesaroni: and I told you that this measure is obviously with respect to the Lebek, and so I can define a density in this way. I I told you that this limit exists almost everywhere, and this limit is equal to one almost everywhere in a in E
01:46:100Annalisa Cesaroni: is equal to 0 almost everywhere outside the
01:50:230Annalisa Cesaroni: this is called the density of the set. So actually, the set of point where this density is equal to one, we call it the the measured theoretic interior of the settee.
02:03:300Annalisa Cesaroni: and the set of point where this density is between 0 and one different from 0, and different from one is for us the so-called measure theoretic boundary. But I want to come back to the definition of this density, because this definition is
02:20:970Annalisa Cesaroni: something which is more general. Which also for other for all the other measure say, and I want to say something about this, because actually, we will need at some point theorem. So
02:36:220Annalisa Cesaroni: so I have to recall something about the notion of dispenciation of measures, and
02:44:290Annalisa Cesaroni: so
02:47:290Annalisa Cesaroni: differentiation of measures. That may be
02:52:410Annalisa Cesaroni: the ones of you that's already followed the course of measure theory already know. Okay, so differentiation of measure. So let's consider, this is a theorem.
03:07:180Annalisa Cesaroni: Maybe it's called Bezikovich. I don't know. So the theorem says the following, let mu
03:15:900Annalisa Cesaroni: be a random measure
03:18:300Annalisa Cesaroni: on our end. Let's stay on the radon case.
03:22:330Annalisa Cesaroni: and
03:24:950Annalisa Cesaroni: we can also stand in some sense. But let's stay on the rabbit case. Let me measure, and let's
03:34:130Annalisa Cesaroni: let
03:40:500Annalisa Cesaroni: such. That mu is absolutely continuous. With respect to Lebesgue, this is Lebesgue
03:48:480Annalisa Cesaroni: mu is absolutely continuous with respect to the bank. Okay.
03:52:280Annalisa Cesaroni: then, what does it mean? It means that every set which has measured 0 with respect to Lebega has also measured 0. With respect to Mu, okay.
04:01:770Annalisa Cesaroni: the set of measured 0 in the bag are also of measured 0. With respect to mu, okay, then there exist.
04:09:360Annalisa Cesaroni: then, for almost
04:12:360Annalisa Cesaroni: every x, almost in the back sense.
04:15:530Annalisa Cesaroni: and also in new sense, because they are absolutely mu, is absolutely continuous with respect to Lebesgue, but almost everywhere, in the sense of Lebesgue, for almost every X in Iran there exists finite
04:29:770Annalisa Cesaroni: the limit, as R. Goes to 0, plus of what of the measure of the
04:37:250Annalisa Cesaroni: the measure with respect to Mu of the Ball Center attacks of Rajusra are.
04:43:600Annalisa Cesaroni: I'm taking the open ball, but open ball and close ball is the same as the same measure, because new is absolutely continue, is a.
04:54:560Annalisa Cesaroni: It's absolutely obvious with respect to the back now, so the the boundary of the set is measured. 0. Okay, the boundary of every set is measured 0 over over the measure of the ball
05:07:270Annalisa Cesaroni: in the back sense
05:09:550Annalisa Cesaroni: which is here we cannot also we can also eliminate the the point because the the bag is invariant by translational. So it is the measure of this book.
05:24:170Annalisa Cesaroni: and the x is finite, and I call it F of x, and I can
05:29:460Annalisa Cesaroni: for almost every X
05:32:20Annalisa Cesaroni: call it F of X, and also F is
05:39:75Annalisa Cesaroni: measurable
05:42:180Annalisa Cesaroni: is measurable
05:45:900Annalisa Cesaroni: in the sense of Lebek. Okay.
05:49:130Annalisa Cesaroni: is a function of X, okay, for every XX is here. The center of the border.
05:56:220Annalisa Cesaroni: If you want, you can put also here the center of the ball. But
06:00:20Annalisa Cesaroni: there's not the here. I don't see the center of the ball. The measure of a ball is just the measure of a ball independently in the back sense. Okay.
06:08:770Annalisa Cesaroni: there's some problem and the same the same theorem also.
06:14:100Annalisa Cesaroni: For if I consider a new radon and new radon
06:20:870Annalisa Cesaroni: to rado measure such that mu is absolutely continuous with respect to mu to Nu, then the same also.
06:27:940Annalisa Cesaroni: when I take the the the
06:31:750Annalisa Cesaroni: limit of the the the ratio of the
06:40:270Annalisa Cesaroni: if I take instead of mu rather, and if I take mu and mu both rather measure. And I take mu absolutely continuous. With respect to Mu, what does mean? It means that the set of measure 0 with respect to Mu, are also of measure 0. With respect to Mu, okay.
06:59:30Annalisa Cesaroni: so
07:00:220Annalisa Cesaroni: the the blue part is a generalization for a given generic random measure. Here I have the case in which Nu is equal to the back, which is a random measure. Okay.
07:10:770Annalisa Cesaroni: this is a generalization, and they add that for almost every X. With respect to Nu, and almost every there exists this limit and is measurable with respect to Nu. Okay.
07:26:413Annalisa Cesaroni: so let's stay on the case of new equal to. But actually, the fact that 12 is not is not the point of this theory. Okay, it's not the point of this theorem. So
07:40:540Annalisa Cesaroni: this app is called is.
07:45:450Annalisa Cesaroni: in some sense we are. We are. And I'm sorry I'm putting the absolute value here. And it's not. We don't need.
07:53:660Annalisa Cesaroni: Okay, obviously, I was thinking to look like
08:05:210Annalisa Cesaroni: and so this is a sort of differentiation of this measure with respect to the back, in the sense that I see I'm taking a measure of 4 with value shrinking to 0.
08:18:930Annalisa Cesaroni: And I, considering the limit of the ratio of this measure with respect to the duration of the measure of the store with respect to the the measure of the ball width of the bed. Okay, and if this limit exists.
08:32:919Annalisa Cesaroni: I call it the density, the density of dysfunction.
08:36:140Annalisa Cesaroni: And
08:38:799Annalisa Cesaroni: and actually
08:40:289Annalisa Cesaroni: actually is also the density is measurable. And it is also the density of mu which appears
08:48:387Annalisa Cesaroni: in theory. Okay.
08:51:900Annalisa Cesaroni: so let's let's see a corollary very important corollary of this theorem, which is called also
09:03:790Annalisa Cesaroni: which is a theorem by itself, so it can be proved also without passing through this differentiation measure, which is called the
09:11:810Annalisa Cesaroni: vitaly lebek theorem.
09:14:410Annalisa Cesaroni: or
09:15:470Annalisa Cesaroni: sometime
09:18:360Annalisa Cesaroni: just the Lebek theorem.
09:20:300Annalisa Cesaroni: sometimes without vitaly
09:22:640Annalisa Cesaroni: with the bacterium, which is the following, let
09:27:350Annalisa Cesaroni: S in l. 1 lock
09:29:860Annalisa Cesaroni: over in
09:31:980Annalisa Cesaroni: l. 1 lock, what does it mean? It means that it is measurable. F is measurable, and it is it has a
09:43:460Annalisa Cesaroni: It is in l. 1 for every if in l 1 of K for every K compact inside. Rn, okay.
09:50:710Annalisa Cesaroni: so it means that F
09:53:70Annalisa Cesaroni: is in l 1 of K for every K contact inside. Rn, okay, l. 1 lock means this, then.
10:01:400Annalisa Cesaroni: if F is in one lot directly for almost every, for almost every x in a M.
10:08:450Annalisa Cesaroni: In Iran there exist.
10:10:550Annalisa Cesaroni: and we have that to say, F of X is equal to limit as R goes to 0, plus of what of one over the measure of the ball of center 0 radius R. With respect to the Bag
10:23:730Annalisa Cesaroni: Times, the integral in Bxr of F of y. Dy.
10:34:100Annalisa Cesaroni: Okay, everything. So this, this Internet is what defined because
10:40:550Annalisa Cesaroni: the ball, the closure of a ball is a compact set. Okay in a re, obviously, so, this integral is what defined.
10:48:560Annalisa Cesaroni: Okay.
10:49:640Annalisa Cesaroni: because F is in l 1 law for every x. So I have to take the integral with respect to F restricted to the ball, and then I divide by the measure of the ball. So it is the mean
11:03:250Annalisa Cesaroni: and taking them in. Okay, the integral of F,
11:07:391Annalisa Cesaroni: and I have that for almost every X. This exists. Why, this is a corollary of the previous theorem. Why, this is a coronary, because, if I define
11:20:90Annalisa Cesaroni: this is a corollary of the previous theorem
11:26:00Annalisa Cesaroni: oh.
11:31:250Annalisa Cesaroni: banner.
11:36:450Annalisa Cesaroni: it's a corollary of the previous
11:43:910Annalisa Cesaroni: been corollary of the previous because
11:48:770Annalisa Cesaroni: of the previous theorem.
11:52:250Annalisa Cesaroni: because if I define this, the the measure new I take F in, and one lock.
11:59:70Annalisa Cesaroni: If F is in l. 1 lock over N.
12:03:30Annalisa Cesaroni: And I take the measure, Nu of a defined in this way the integral of a of Fy dy.
12:13:180Annalisa Cesaroni: So for every a moral.
12:16:110Annalisa Cesaroni: I define new of A in this way.
12:19:960Annalisa Cesaroni: Okay.
12:23:200Annalisa Cesaroni: S, positive. Say.
12:25:940Annalisa Cesaroni: if it is negative or okay.
12:29:110Annalisa Cesaroni: if F is, let's stay on F positive. If F is positive, Nu is a positive, for every a new is positive nu of A is positive, then actually.
12:41:224Annalisa Cesaroni: this is absolutely continuous. With respect to Lebega, new Israel.
12:46:490Annalisa Cesaroni: wise Ravon
12:49:110Annalisa Cesaroni: new is about measure. One can find and want to show that it is a boral measure. Okay, one has to use water I have. We have to use the the
13:01:280Annalisa Cesaroni: the property of telepag integral for all this kind of things.
13:05:380Annalisa Cesaroni: and if I have is in l. 1 lock. It means that the if I consider the measure of a compact set, the measure of a compact set is finite, because F is in l 1 log. Okay.
13:19:830Annalisa Cesaroni: okay, so this is the case of positive function. Then we can. Okay, for so for positive function, it is
13:31:430Annalisa Cesaroni: It is obvious that
13:33:830Annalisa Cesaroni: this theorem here is actually exactly
13:38:440Annalisa Cesaroni: this theorem here. Okay?
13:40:810Annalisa Cesaroni: Because in sorry. Mu, I call it mu of a.
13:47:850Annalisa Cesaroni: because mu of the ball is what mu of the ball of center X radius R is actually
13:55:839Annalisa Cesaroni: this. This quantity here is actually this quantity here.
14:00:750Annalisa Cesaroni: Okay, by definition, and then I divide by the measure of the ball. And it is exactly the same. Okay.
14:08:800Annalisa Cesaroni: but
14:11:240Annalisa Cesaroni: okay. And then we can also treat the case in which F is not positive. So writing it as the difference of the positive and the negative part, etcetera. Okay.
14:25:160Annalisa Cesaroni: So in some sense it is a corollary, and actually.
14:33:110Annalisa Cesaroni: And actually, since the previous theorem is also true for mu and mu both random measure, then it is also true. The previous theorem. If here, I don't take FL. 1 log with respect to the bag, if I take FL. 1 log with respect to a random measure, Mu.
14:54:850Annalisa Cesaroni: okay. So if New Israel
14:59:710Annalisa Cesaroni: and the F
15:01:230Annalisa Cesaroni: is in l 1 lock.
15:04:124Annalisa Cesaroni: With respect to new, say, let's let's write like this, and then F of X is equal to the limit.
15:12:190Annalisa Cesaroni: for almost
15:14:870Annalisa Cesaroni: every x, almost every in new sense.
15:18:710Annalisa Cesaroni: say, say, anyway, is the limit, as R goes to 0 plus of
15:23:940Annalisa Cesaroni: one over
15:28:640Annalisa Cesaroni: mu of Bxr
15:32:756Annalisa Cesaroni: integral over BXR.
15:35:970Annalisa Cesaroni: F. Of YD, mu of y.
15:39:200Annalisa Cesaroni: Okay.
15:41:310Annalisa Cesaroni: the the same. Also, when I don't consider the integration with respect to the bank. But the integration with respect to
15:49:190Annalisa Cesaroni: okay.
15:50:260Annalisa Cesaroni: so actually, maybe there is an exercise. If F is positive and U is Radon.
15:58:220Annalisa Cesaroni: then the the
16:01:980Annalisa Cesaroni: F is positive at the integral, and l. 1 lock. With respect to
16:07:580Annalisa Cesaroni: FL. 1 lock
16:11:320Annalisa Cesaroni: with respect to new.
16:12:900Annalisa Cesaroni: So the the measure mu of a, the finance, the integral of a of Foy.
16:18:720Annalisa Cesaroni: the new of y.
16:21:120Annalisa Cesaroni: Maybe this was an exercise of this kind is a measure which is absolutely continuous with respect to new. And it is also other. Okay.
16:31:160Annalisa Cesaroni: actually, exactly.
16:32:970Annalisa Cesaroni: It makes sense, some sense what I'm seeing. And there is something strange, because I'm seeing
16:39:660Annalisa Cesaroni: if there is something strange. Please stop me and ask me.
16:44:800Annalisa Cesaroni: Okay. So actually, it's just saying that
16:48:270Annalisa Cesaroni: the previously bacterium
16:50:370Annalisa Cesaroni: that I am, I think that you already know.
16:55:51Annalisa Cesaroni: Take another measure, random measure and integrate with respect to random.
17:04:82Annalisa Cesaroni: Okay? And
17:08:109Annalisa Cesaroni: these this in this corollary. Can. Also, we can state like this. We can state in different ways so we can state.
17:21:400Annalisa Cesaroni: say, if F is in lp, lock over N
17:26:690Annalisa Cesaroni: for P in
17:28:430Annalisa Cesaroni: between one and plus infinity. I can also say that the limit as R goes to 0
17:36:730Annalisa Cesaroni: plus
17:38:640Annalisa Cesaroni: of one over B
17:40:970Annalisa Cesaroni: at 0 h. Let's
17:44:90Annalisa Cesaroni: the ball.
17:45:530Annalisa Cesaroni: Oh.
17:47:630Annalisa Cesaroni: bxr, F of y minus f of x to the power P in the y is equal to 0 for almost every x
17:57:510Annalisa Cesaroni: in Iran once that I have corollary of the corollary disease.
18:04:230Annalisa Cesaroni: Let's see
18:06:320Annalisa Cesaroni: one that I have that theorem. I have, that if I have a function which is in Lp lock.
18:13:30Annalisa Cesaroni: So say, let's start with F in l. 1 lock. If I have F in l 1 lock. Actually, I have that if F is in l. 1 lock.
18:24:660Annalisa Cesaroni: I have that
18:26:960Annalisa Cesaroni: the integral over the ball of F of y, minus F of X modulus converts to F,
18:33:60Annalisa Cesaroni: divided by the measure of the ball
18:35:870Annalisa Cesaroni: converts to 0 as a r goes to 0
18:39:240Annalisa Cesaroni: as the radius of the ball goes to 0, so for P. Equal to one is the force, the generalization of the previous. The previous statement I take F of X on the other side. Okay, I take it inside the integral. And then I use the property of the Lebesgue integral. Okay.
18:59:970Annalisa Cesaroni: so this is quite immediate as A,
19:04:280Annalisa Cesaroni: and then the same can be also done for F in lp, log. What does mean? Lp, log means that F to the power P is integrable, as finite, integral in every compact set in our end.
19:17:180Annalisa Cesaroni: And so you can also see.
19:19:50Annalisa Cesaroni: Okay. But this can. All all these kind of things can be reduced by the 1st theorem.
19:25:740Annalisa Cesaroni: which is the differentiation of measure, and the 1st theorem says, in which way I can also, I can define the density of an absolutely continuous measure with respect to another. Okay, these are a way in which I can construct the density. Okay.
19:46:260Annalisa Cesaroni: did they execute this one? No.
19:51:176Annalisa Cesaroni: obviously, if you, if new is okay
19:57:470Annalisa Cesaroni: and okay. And
20:03:100Annalisa Cesaroni: And for example, if view is the the measure that they define on Wednesday, which is the measure okay associated to a set to the density is the density of the set. Okay, in the generalized sense. Okay? So
20:19:791Annalisa Cesaroni: how to prove this theorem?
20:21:900Annalisa Cesaroni: How to prove this theorem? One can prove either this theorem here or the other theorem. Here the Lebeck theorem can be proved directly.
20:30:860Annalisa Cesaroni: Both this theorem are based. The proof of both this theorem are based on
20:37:615Annalisa Cesaroni: very important, very important results in dramatic measure theory, which are called the covering theorems, which are the vitale and the Besikovich covering theorem. And since we are going to use sometimes. This, I'm going to to
20:53:250Annalisa Cesaroni: state this theorem in a reduced form. But not to prove this theorem, you can find the proof of this theorem, if you are interested in the book by Evan Sengariabi. But I'm not going to prove this theorem. It is more for a
21:08:440Annalisa Cesaroni: for a course of measure of theory, say
21:14:620Annalisa Cesaroni: so.
21:18:984Annalisa Cesaroni: this, the proof of the the proof
21:23:830Annalisa Cesaroni: of this theorem
21:29:170Annalisa Cesaroni: are based.
21:31:00Annalisa Cesaroni: And then with this week, more or less, we are going to finish with the part of dramatic micro theory and then pass to the
21:41:287Annalisa Cesaroni: got it.
21:46:440Annalisa Cesaroni: Okay.
21:47:920Annalisa Cesaroni: more can stop in a soroma.
22:17:580Annalisa Cesaroni: See?
22:18:690Annalisa Cesaroni: Yeah, bye, bye?
22:21:553Annalisa Cesaroni: Oh, no.
22:30:507Annalisa Cesaroni: the notes.
22:38:560Annalisa Cesaroni: So let's go to the platform. And the
22:45:210Annalisa Cesaroni: so the problem of this was saying on this result, which are called theorems, which are very important theorem. The 1st covid theorem, is due to the time.
22:59:70Annalisa Cesaroni: was a Italian mathematician, and it dates back to the beginning of the 19th century. The second calling theorem is a generalization of the Italian poetic theorem.
23:09:440Annalisa Cesaroni: because vitaly correct measure. And then there is a generalization by Vizikkovich with a Russian mathematician, and it is a generalization to general Radon measure. Okay? So the 1st is Vitali, and then there is Desikkovich.
23:30:200Annalisa Cesaroni: and this guitar is the beginning of 19th century.
23:35:230Annalisa Cesaroni: So 19 8, something like this. And basically is 1940, 48.
23:43:60Annalisa Cesaroni: And let's just take directly for the sequence.
23:46:790Annalisa Cesaroni: So basically, just
23:49:960Annalisa Cesaroni: well, good evening.
23:53:820Annalisa Cesaroni: You know what else.
23:56:460Annalisa Cesaroni: There are different ways of stating this theorem I'm just taking in the version that I'm going to use directly. So actually, if one is precise. This is a total of the this, the 2 basic of each theorem. But okay.
24:12:570Annalisa Cesaroni: busy connection.
24:17:160Annalisa Cesaroni: So forties around forties and Italy.
24:30:470Annalisa Cesaroni: so vital is just for event.
24:33:710Annalisa Cesaroni: and actually having just in the back once can say something, one can say something more. But we are not going to deliver this.
24:44:650Annalisa Cesaroni: So actually, the Italians say something more. But
24:53:425Annalisa Cesaroni: okay, so let the mute.
25:00:00Annalisa Cesaroni: when rather okay, positive measure on our end.
25:13:110Annalisa Cesaroni: And then the
25:15:920Annalisa Cesaroni: we can see the F.
25:18:920Annalisa Cesaroni: A connection
25:24:390Annalisa Cesaroni: of close the board.
25:30:960Annalisa Cesaroni: So I close the board close bowl. And this connection is not degenerate
25:37:640Annalisa Cesaroni: in the sense that
25:40:240Annalisa Cesaroni: all values are positive
25:43:550Annalisa Cesaroni: already are positive.
25:47:225Annalisa Cesaroni: Okay?
25:49:460Annalisa Cesaroni: And they consider a
25:52:180Annalisa Cesaroni: the sector of the centers.
25:54:790Annalisa Cesaroni: Oh, boy.
25:59:570Annalisa Cesaroni: and assume that I call it New New Way less than passing
26:10:850Annalisa Cesaroni: fine, and I assume also that this collection is fine.
26:17:420Annalisa Cesaroni: His collection is fine in the sense that they implement and assume also that they implement.
26:26:620Annalisa Cesaroni: let's say.
26:28:220Annalisa Cesaroni: for every center email
26:32:690Annalisa Cesaroni: of Viragua.
26:34:650Annalisa Cesaroni: such that the door of center A and Rajastar closed.
26:39:180Annalisa Cesaroni: Easy enough is equal to 0.
26:42:640Annalisa Cesaroni: Okay? So for every day
26:45:870Annalisa Cesaroni: I can, I consider also
26:48:890Annalisa Cesaroni: that's a collection of those ball. Okay.
26:52:751Annalisa Cesaroni: I consider a the center of all the voice.
26:56:40Annalisa Cesaroni: And I consider, and I assume
26:59:210Annalisa Cesaroni: that for every center
27:01:940Annalisa Cesaroni: what I just sent them.
27:03:700Annalisa Cesaroni: the interim of the radium, the radium, such that the ball of center Aradius, is an element of this family of balls is equal to 0,
27:16:170Annalisa Cesaroni: so I can find.
27:18:750Annalisa Cesaroni: and also overrides as more as I want.
27:22:130Annalisa Cesaroni: Sorry inside with a inside the collection app.
27:27:260Annalisa Cesaroni: then.
27:28:530Annalisa Cesaroni: then.
27:30:330Annalisa Cesaroni: for every you
27:32:880Annalisa Cesaroni: not an open set.
27:37:190Annalisa Cesaroni: open sector that exist? What is the problem with this collection of board?
27:43:350Annalisa Cesaroni: The problem is is that this collection of board is in general accountable.
27:48:400Annalisa Cesaroni: So the point is that it's we have to work with accountable set. And
27:54:690Annalisa Cesaroni: so for every you in our open set that exist.
28:01:166Annalisa Cesaroni: Good thing next
28:05:870Annalisa Cesaroni: G countable.
28:08:980Annalisa Cesaroni: comfortable.
28:10:570Annalisa Cesaroni: friendly.
28:12:940Annalisa Cesaroni: Oh, close board of of course. Born in action.
28:21:640Annalisa Cesaroni: please join.
28:27:300Annalisa Cesaroni: Okay, so G is a subset is a sub family accountable sub family of the boards in F made of disjoint board
28:41:110Annalisa Cesaroni: this joint.
28:42:690Annalisa Cesaroni: Okay.
28:44:960Annalisa Cesaroni: So G. There exists G. Countable family of disjoint Cross board in F. So the point is that they are. The 1st point is, they are countable. The second point they are disjoint
28:58:640Annalisa Cesaroni: such that
29:04:565Annalisa Cesaroni: okay
29:06:340Annalisa Cesaroni: is covered by this this union of board, so
29:11:180Annalisa Cesaroni: it measure sense
29:12:640Annalisa Cesaroni: minus the Union. In G of
29:20:490Annalisa Cesaroni: this ball is a set
29:23:360Annalisa Cesaroni: with new measure equal to 0.
29:28:370Annalisa Cesaroni: So a intersected group is covered.
29:33:190Annalisa Cesaroni: Bye.
29:35:40Annalisa Cesaroni: are you New York?
29:43:65Annalisa Cesaroni: This joint
29:47:180Annalisa Cesaroni: balls in
29:55:840Annalisa Cesaroni: another way in which I can state this.
30:01:930Annalisa Cesaroni: is the following, say.
30:11:970Annalisa Cesaroni: because it is a bit a bit strange, you know. I have a collection of call.
30:18:360Annalisa Cesaroni: and they came home. You intercepted the the family of the sector, the board. So
30:26:820Annalisa Cesaroni: the way in which way we have to think of it is the following, we start.
30:32:350Annalisa Cesaroni: take off the case.
30:34:560Annalisa Cesaroni: If we took.
30:35:960Annalisa Cesaroni: I fix you open.
30:38:850Annalisa Cesaroni: open.
30:40:502Annalisa Cesaroni: say, a measure of you, financial.
30:43:740Annalisa Cesaroni: open and bounded.
30:45:610Annalisa Cesaroni: Let's fix an open set like this. Okay, and let's fix that a positive
30:51:710Annalisa Cesaroni: and consider, or
30:55:310Annalisa Cesaroni: like this say.
30:57:170Annalisa Cesaroni: or more
31:02:210Annalisa Cesaroni: All the closure on board
31:05:620Annalisa Cesaroni: contain new U is open where X is new, and R is less or equal than Delta.
31:14:30Annalisa Cesaroni: Let's take this set F,
31:17:140Annalisa Cesaroni: all the ball we center in you values less than that which are inside you.
31:24:690Annalisa Cesaroni: Okay?
31:28:300Annalisa Cesaroni: So actually, if if you use this one, and this effect is very nearby to the boundary. I will take just maybe a distance less than that from the boundary. I will take just the ball center at the text with reduce strictly less than that, that's strictly less than the distance one. And just say so. Think of this case.
31:52:950Annalisa Cesaroni: And so what is what is saying? This theorem, this theorem says, that there exists a countable family of balls contained this joint board contained in you
32:06:670Annalisa Cesaroni: of Radus less than Delta which covers you.
32:10:80Annalisa Cesaroni: Okay, so did you run, says.
32:15:810Annalisa Cesaroni: but Brexit, accountable
32:19:860Annalisa Cesaroni: for a minute.
32:24:110Annalisa Cesaroni: Say, Xn in U. Rn. Less than that such that the goal of Center Xn
32:34:230Annalisa Cesaroni: have disjoint
32:40:140Annalisa Cesaroni: one from the other.
32:45:970Annalisa Cesaroni: the closure, the closure of disjoint, one from the other.
32:49:950Annalisa Cesaroni: Okay.
32:51:400Annalisa Cesaroni: in the sense that and such that you is contain. Say, the the new of you, minus the union
33:03:220Annalisa Cesaroni: of the Xn Rn
33:08:250Annalisa Cesaroni: is equal to 0. So U is over by this. Okay.
33:13:240Annalisa Cesaroni: I'm saying this. So what does it here? I'm saying that safe.
33:19:640Annalisa Cesaroni: Let's write the B of Xn Rn. Intersect B of XM.
33:28:840Annalisa Cesaroni: Alright
33:30:630Annalisa Cesaroni: supposed to be up to set.
33:33:730Annalisa Cesaroni: Okay.
33:38:860Annalisa Cesaroni: so
33:40:780Annalisa Cesaroni: this is actually the year 2 minus 12 of this year. This is the tier, and if new is
33:52:610Annalisa Cesaroni: a rather measure on our end. So let's into the case of the bank.
33:58:280Annalisa Cesaroni: And so this family, I hope, I said, I take an open set. I consider this family. This family, obviously.
34:07:242Annalisa Cesaroni: satisfies this because new, the center of it of the boards are just the set tube that I I
34:14:600Annalisa Cesaroni: I think it would be fine measure. Okay?
34:18:580Annalisa Cesaroni: And also it satisfies also this, okay.
34:22:880Annalisa Cesaroni: because I'm taking all the board here with center new, and we reduce less than equal delta
34:30:510Annalisa Cesaroni: for the board which stays inside you, so that has to be strictly less than the distance between X and the boundary of you.
34:38:100Annalisa Cesaroni: Okay, so this is the way in which this theory is used.
34:42:929Annalisa Cesaroni: and actually.
34:47:00Annalisa Cesaroni: just to give an idea.
34:50:00Annalisa Cesaroni: if some of you never seen the bacterium, the 2 of the bacterium of the product differentiation theorem. Just to have an idea of
35:01:600Annalisa Cesaroni: why, this year I'm not missing
35:04:477Annalisa Cesaroni: in the statement above. We have.
35:08:790Annalisa Cesaroni: hey? Intersect mute?
35:11:820Annalisa Cesaroni: Yes, because in this case a is equal to B,
35:16:267Annalisa Cesaroni: because, yeah, this is just an hour.
35:22:230Annalisa Cesaroni: The the 1st item is the more general one. No.
35:26:340Annalisa Cesaroni: And actually, one can think to this case.
35:29:660Annalisa Cesaroni: one can do to fix the set of sites so- so the fix to fix.
35:36:210Annalisa Cesaroni: Oh.
35:37:520Annalisa Cesaroni: so actually, here in the theorem. I have a generic collection of ports
35:44:280Annalisa Cesaroni: and the generic family of
35:47:922Annalisa Cesaroni: centers of Boston with the net measure. And then I I can cover you intersect today with this disjoint. But now
36:01:476Annalisa Cesaroni: the point is that actually, what is what?
36:05:100Annalisa Cesaroni: So you in this way, maybe. Okay. I could also have a stated directly in this way. But
36:14:660Annalisa Cesaroni: okay, actually, the point is, the is the one I want to cover an open set.
36:20:330Annalisa Cesaroni: bounded open, set with a fine and accountable union of disjoined balls over some data suggested by me. Okay, so what I'm doing, I'm taking a collection of ball such that the centers coincide the center of the balls will coincide with you.
36:42:80Annalisa Cesaroni: So there a intersected U is equal to U.
36:46:190Annalisa Cesaroni: I want to cover, you know. So I take a family such that all the center are inside you
36:52:609Annalisa Cesaroni: and
36:53:200Annalisa Cesaroni: and the ball, and I consider all the in this family. All the closure of ball with center in radus are less than delta, such that the closure of the ball is inside you itself.
37:09:240Annalisa Cesaroni: So in this way I can apply this theorem, because F satisfies all, all the condition in the theorem, and I can find out
37:19:720Annalisa Cesaroni: of the open statue by a disjoint.
37:23:230Annalisa Cesaroni: countable union of boards
37:25:870Annalisa Cesaroni: of small values.
37:34:310Annalisa Cesaroni: So just take me and just fix it.
37:38:780Annalisa Cesaroni: does it?
37:39:950Annalisa Cesaroni: We have to put the centers is another problem. But
37:46:190Annalisa Cesaroni: let's come cold up
37:48:940Annalisa Cesaroni: with leaving nothing outside. Nothing in the measure sense with this joint goal with center in.
37:58:270Annalisa Cesaroni: So actually, this is a way in which I can apply this theorem, say a intersected you with something strange, and the- the center the open set. So
38:09:800Annalisa Cesaroni: I fixed the open set that I want to cover.
38:13:290Annalisa Cesaroni: and I I find out a family. Of course we sent it.
38:17:900Annalisa Cesaroni: Who centers up coincide with you.
38:21:770Annalisa Cesaroni: Okay, so if you prefer, you can state
38:25:820Annalisa Cesaroni: directly like this.
38:27:790Annalisa Cesaroni: If it is less.
38:31:860Annalisa Cesaroni: This, this can be seen. This can be thought as an equivalent statement of the theory. Maybe the other was a bit misleading. I don't know.
38:42:670Annalisa Cesaroni: Oh.
38:43:820Annalisa Cesaroni: okay.
38:50:885Annalisa Cesaroni: And
38:54:170Annalisa Cesaroni: and that show you.
38:59:750Annalisa Cesaroni: And actually, the point is that we can cover set open, set with the disjoint accountable both.
39:09:870Annalisa Cesaroni: and the the fact that they have disjoined.
39:13:470Annalisa Cesaroni: And the accountable is useful in the when we are going to apply measure some theories which I want to.
39:24:270Annalisa Cesaroni: For example, instead of considering the measure of you. I can consider this measure, which will be the the sum of the measure of this book.
39:34:540Annalisa Cesaroni: Okay.
39:35:730Annalisa Cesaroni: because they are all disjoint, you know, as I mentioned. So
39:45:640Annalisa Cesaroni: and
39:47:680Annalisa Cesaroni: okay, and this theorem will be very useful in a lot of different
39:56:30Annalisa Cesaroni: things. But I don't know. If
40:08:00Annalisa Cesaroni: so, let's let's see very quickly. This is not a theorem that I'm going to ask or to put in the program, but the approval that I'm going to put in the program.
40:19:20Annalisa Cesaroni: But just to understand why we need this covention to prove the differentiation of measure. So let's fix
40:28:110Annalisa Cesaroni: new.
40:32:440Annalisa Cesaroni: And I want to prove that the limit
40:36:64Annalisa Cesaroni: goes to 0, plus
40:38:220Annalisa Cesaroni: of the mute
40:44:910Annalisa Cesaroni: over to the measure of the board with respect to the bank, is
40:53:320Annalisa Cesaroni: clients
40:55:340Annalisa Cesaroni: for almost 10 x, and it's also measurable.
41:01:920Annalisa Cesaroni: So let's see in which way I can do this. Okay.
41:06:470Annalisa Cesaroni: so I'm just
41:08:945Annalisa Cesaroni: which way I can book this just giving a very brief sketch very brief sketch. So
41:20:880Annalisa Cesaroni: so
41:22:800Annalisa Cesaroni: let's do like this. So I have to reduce the contact center.
41:27:840Annalisa Cesaroni: Oh.
41:30:870Annalisa Cesaroni: in order to
41:34:450Annalisa Cesaroni: got it
41:35:940Annalisa Cesaroni: okay, to ask them.
41:39:160Annalisa Cesaroni: no fix, Alpha positive and consider
41:45:510Annalisa Cesaroni: a
41:46:890Annalisa Cesaroni: contain in the site correct, such that the link
41:52:10Annalisa Cesaroni: as the Lima
41:57:690Annalisa Cesaroni: the limit link soup as R goes to 0, plus of this quantity.
42:06:370Annalisa Cesaroni: Hold on
42:07:690Annalisa Cesaroni: XR.
42:13:820Annalisa Cesaroni: the reason, alpha.
42:19:620Annalisa Cesaroni: and let a like this.
42:25:00Annalisa Cesaroni: then
42:27:250Annalisa Cesaroni: then actually, one can prove that mu of a
42:34:670Annalisa Cesaroni: is less than alpha, the back measure of a
42:38:960Annalisa Cesaroni: hey?
42:47:340Annalisa Cesaroni: But it's not important.
42:49:530Annalisa Cesaroni: So this link should exist for us.
42:53:950Annalisa Cesaroni: I take X such that this limb super is bigger or equal than Alpha.
43:00:555Annalisa Cesaroni: Okay?
43:03:960Annalisa Cesaroni: So in China, this limit depends on the sequences. So I take for every sequence is going to film. I take the Supreme.
43:18:40Annalisa Cesaroni: and then
43:29:300Annalisa Cesaroni: let's reduce to the case in which and consider a inside
43:42:320Annalisa Cesaroni: extra
43:44:806Annalisa Cesaroni: such that. And then we will. We will get rid of this, the complex set.
43:59:80Annalisa Cesaroni: Okay, just to have everything of fine attention. Okay? So let's fix a contractor.
44:06:190Annalisa Cesaroni: And then
44:07:820Annalisa Cesaroni: the reasoning will be for a very compact. And then
44:12:580Annalisa Cesaroni: and then what we are going to do.
44:15:720Annalisa Cesaroni: we are going to consider. So
44:20:364Annalisa Cesaroni: actually, I fixed the
44:22:570Annalisa Cesaroni: you opened.
44:24:790Annalisa Cesaroni: and a company continues
44:27:300Annalisa Cesaroni: such that the measure of U is finite.
44:30:580Annalisa Cesaroni: because A is a subset of K measure find. So it's an open set which contains a
44:39:370Annalisa Cesaroni: okay
44:41:540Annalisa Cesaroni: with the final measure. And then I apply this, you the covering here. But then I have to decide which is the family of both using which I want to cover. Okay.
44:55:110Annalisa Cesaroni: so
44:57:230Annalisa Cesaroni: I take this family, the family of the person.
45:07:210Annalisa Cesaroni: and take the family of the balls like this. The family
45:12:362Annalisa Cesaroni: of close ball always closed, such that this ball is big.
45:18:80Annalisa Cesaroni: A, r with the
45:20:760Annalisa Cesaroni: hey, may
45:24:322Annalisa Cesaroni: we set here? Okay.
45:28:240Annalisa Cesaroni: BAR contains like you, the open set starting
45:34:60Annalisa Cesaroni: okay, and and
45:38:60Annalisa Cesaroni: and such that new
45:44:246Annalisa Cesaroni: sorry mute.
45:46:460Annalisa Cesaroni: Oh, this board!
45:50:710Annalisa Cesaroni: It's less really cool. Then
45:53:160Annalisa Cesaroni: Alpha classic sine
45:57:860Annalisa Cesaroni: the measure of the goal in the back sense.
46:02:960Annalisa Cesaroni: Fix epsilon. So let's call it F. Epsilon.
46:08:520Annalisa Cesaroni: Okay? So I'm taking a family of goals
46:12:660Annalisa Cesaroni: like this. The center are in the center. A, the center are in the set a.
46:19:330Annalisa Cesaroni: The radios are chosen in such a way that this board of center, a. The Radusa closed, is inside the open set.
46:28:910Annalisa Cesaroni: which contains saying that I fixed them, and, moreover, and moreover, the measure
46:36:10Annalisa Cesaroni: Mu is a random measure. The measure of this volts has to be less than Alpha plus silon. Where alpha, is this
46:46:100Annalisa Cesaroni: this positive constant here.
46:50:00Annalisa Cesaroni: alpha plus epsilon times. The limbic measure of the ball.
46:55:250Annalisa Cesaroni: Okay, off center area.
46:58:480Annalisa Cesaroni: close the or open. And okay. R, is R. So actually, this condition.
47:04:660Annalisa Cesaroni: this condition is a condition which implies in particular, that once that I have an arm
47:10:890Annalisa Cesaroni: inside such that these for which this also.
47:16:440Annalisa Cesaroni: Yeah, actually.
47:18:710Annalisa Cesaroni: then, actually.
47:23:980Annalisa Cesaroni: I can take a
47:26:640Annalisa Cesaroni: but it is not completely obvious. But actually, I can take radios as more as I want
47:34:380Annalisa Cesaroni: doing this.
47:36:720Annalisa Cesaroni: He's a
47:39:990Annalisa Cesaroni: so actually, this family F of Epsilon is a family to which I can apply the previous theorem. And then I have what I have that exist a family of this joint pool
47:57:220Annalisa Cesaroni: there exist a family of this, so report I can apply the the
48:02:540Annalisa Cesaroni: I can apply them to you. I apply
48:07:940Annalisa Cesaroni: the covid theorem.
48:12:800Annalisa Cesaroni: and they have what and they have that there exists a family
48:18:280Annalisa Cesaroni: of this joint force.
48:26:100Annalisa Cesaroni: the end
48:27:630Annalisa Cesaroni: inside the you.
48:31:320Annalisa Cesaroni: such that such that
48:33:830Annalisa Cesaroni: the measure of a.
48:36:720Annalisa Cesaroni: because it is a intersected view, but
48:39:820Annalisa Cesaroni: so
48:41:310Annalisa Cesaroni: is that
48:42:800Annalisa Cesaroni: is contained you, and it is the
48:49:210Annalisa Cesaroni: the set of the the center of the force a minus this union of the end
48:57:316Annalisa Cesaroni: is equal to 0.
49:04:650Annalisa Cesaroni: Okay.
49:08:400Annalisa Cesaroni: this is the covering theory applied to new. Okay? A
49:14:400Annalisa Cesaroni: is the set. I fix Alpha, I fix a and I fix an open set which contains a okay.
49:22:650Annalisa Cesaroni: and they apply the previous theorem. As I was writing to the more gender formulation. So here in the A intersected U, but they intersected U is a, because A is inside you. Okay.
49:34:950Annalisa Cesaroni: so this is a minus. And so I have what I have. That mu of A is equal to the sum of mu of the N.
49:45:650Annalisa Cesaroni: See.
49:51:170Annalisa Cesaroni: and
49:59:560Annalisa Cesaroni: yes, and mu of bn is this is less or equal than alpha plus sine
50:07:810Annalisa Cesaroni: times. The sum ren of the measure of the n.
50:13:750Annalisa Cesaroni: The closure of the end is here.
50:17:399Annalisa Cesaroni: Okay. Okay.
50:30:720Annalisa Cesaroni: But then
50:34:340Annalisa Cesaroni: then, what do you have?
50:38:350Annalisa Cesaroni: Oh, yeah. But
50:43:930Annalisa Cesaroni: all the are included inside you.
50:47:920Annalisa Cesaroni: Okay, I've disjoined.
50:50:970Annalisa Cesaroni: And then this is less than equal than the measure than the back measure of you.
50:55:180Annalisa Cesaroni: Okay, so this is alpha plus xion, the back measure of you.
51:00:750Annalisa Cesaroni: So mu of a z lesser equal than alpha plus xi. On the back measure of one.
51:05:900Annalisa Cesaroni: Thanks.
51:07:340Annalisa Cesaroni: then.
51:08:830Annalisa Cesaroni: is also for every you
51:12:620Annalisa Cesaroni: open which contains a.
51:15:580Annalisa Cesaroni: Now the repair measure is actually regular.
51:19:370Annalisa Cesaroni: So actually you have a
51:22:260Annalisa Cesaroni: is less or equal than Alpha plus epsilon times. The Lebeck measure of a, because this equality also mu of A is less or equal than alpha plus epsilon times. The Lebec measure of U for every U open set which contains a
51:39:20Annalisa Cesaroni: but the back measure is regular in the sense that the the back measure of a foreign sector
51:45:550Annalisa Cesaroni: can be approximated by about by the measures of all the open set which contains the given model set. This is for regularity of the rebate measure.
51:56:420Annalisa Cesaroni: and then I let the excitement was to 0,
51:59:630Annalisa Cesaroni: and that's all
52:03:617Annalisa Cesaroni: less sorry, less sorry, less sorry.
52:10:740Annalisa Cesaroni: It's like I I sorry.
52:15:420Annalisa Cesaroni: less suitable, otherwise.
52:23:750Annalisa Cesaroni: Sorry I didn't do the proof for last week. But otherwise, yeah.
52:30:990Annalisa Cesaroni: yeah, that we don't
52:33:650Annalisa Cesaroni: sorry for this.
52:43:686Annalisa Cesaroni: but it's too long.
52:49:540Annalisa Cesaroni: So let's make a break, and then continue.
53:03:920Annalisa Cesaroni: we need less or equals. Yes, because it is sorry.
53:09:620Annalisa Cesaroni: Let's see.
53:14:710Annalisa Cesaroni: No.
53:21:460Annalisa Cesaroni: we need less
53:24:140Annalisa Cesaroni: see.
53:25:410Annalisa Cesaroni: So I'm sorry for this.
53:27:900Annalisa Cesaroni: So
53:36:120Annalisa Cesaroni: sorry if you're just going to let Alpha 0.
53:45:10Annalisa Cesaroni: What? Some?
53:47:104Annalisa Cesaroni: Why, take what role is Epsilon survey?
53:55:01Annalisa Cesaroni: Okay, this family is.
54:08:210Annalisa Cesaroni: Oh.
54:09:840Annalisa Cesaroni: to be sure that this family meets the complication.
54:15:830Annalisa Cesaroni: Because okay.
54:22:235Annalisa Cesaroni: yes, for Alpha directly, because it is not fine cover.
54:33:930Annalisa Cesaroni: Let's see.
54:35:331Annalisa Cesaroni: Okay. But the idea is that actually, one has to use a program in the sense that, okay?
54:44:20Annalisa Cesaroni: And then, yeah, sorry for this.
54:48:210Annalisa Cesaroni: we will continue later after the small break, and then we will see actually that also the other inequality that I had written before also.
54:59:80Annalisa Cesaroni: So let's make a break now.
55:01:840Annalisa Cesaroni: and let's see if
55:05:260Annalisa Cesaroni: I will also write the notes.
55:11:650Annalisa Cesaroni: Yes, the point is that it is not fine. With that oops. Sorry.
55:19:770Annalisa Cesaroni: Let me just
55:41:190Annalisa Cesaroni: tend to
55:44:830Annalisa Cesaroni: news, is it? Doesn't recover.
55:49:270Annalisa Cesaroni: I had a question about. So we started the today's lecture by defining, see? So so I got a little confused. This was a definition. No, no, a theorem. This was a theorem. Okay, but we're calling
56:13:753Annalisa Cesaroni: this. The definition is that okay? The theorem says that there's an implicitly defined F, yes, okay, okay, which is the best. Yes, the definition says that the theorem says that for almost every x there is finite this limit.
56:31:490Annalisa Cesaroni: So there exists an F satisfying this equality. Okay? And then I call it to the density felt like. At 1st I interpreted it as a theorem, and then, when you called it the density. I was like, maybe it's the definition. Okay, so these are all.
56:52:790Annalisa Cesaroni: And in some ways, if if our measure is positive, these are all equivalent. Right? Yes, yes, okay. If F is positive, these are equivalent, because
57:04:570Annalisa Cesaroni: because this is exactly this. Yes, exactly this. And also if F is not positive, I can extend by positive part negative part of a measure, negative part. And so in the denominator here, this would be the yes.
58:08:524Annalisa Cesaroni: So let's start again.
58:20:612Annalisa Cesaroni: Sorry for the.
58:24:904Annalisa Cesaroni: But actually, the point is that actually the the idea is the following.
58:38:360Annalisa Cesaroni: it is the following.
58:40:550Annalisa Cesaroni: that
58:42:940Annalisa Cesaroni: good
58:46:220Annalisa Cesaroni: are we more or less, as I show you in a very confused way, then the point is the following one. The point is this, one picks Alpha positive. We take a inside the set of point.
59:01:670Annalisa Cesaroni: If it's okay, a contact in our end.
59:06:486Annalisa Cesaroni: such that they leave
59:09:770Annalisa Cesaroni: as R goes to 0 of Mu of Bx, R.
59:16:585Annalisa Cesaroni: Over
59:17:823Annalisa Cesaroni: the back measure of the board is, remains less than Alpha.
59:24:490Annalisa Cesaroni: and then one proof that mu of a is less frequent than Alpha times. The measure of a this should be more or less, and the proof of this is done more or less, as I showed you by means of the covering theorem of Besikovych. Okay, the idea is, find a correct family time family of covering moles, and then extract
59:51:30Annalisa Cesaroni: a family as a family which is on board which are disjoint and comfortable, which covers a, and so I can use the fact that the measure is
00:05:290Annalisa Cesaroni: The measure is sigma finite. So these balls are countable is an accountable number, and are destroyed.
00:13:930Annalisa Cesaroni: and coercing the in the measured sense, all the sake
00:19:140Annalisa Cesaroni: and more. And so
00:21:640Annalisa Cesaroni: all these wall are, moreover, disjoint, and containing an open set queue, and then
00:27:234Annalisa Cesaroni: I have these.
00:31:620Annalisa Cesaroni: it's fine, and I can.
00:34:700Annalisa Cesaroni: I can. I can use the fact that the back measure is
00:39:890Annalisa Cesaroni: is regular. Then actually, also, I can find can find.
00:48:200Annalisa Cesaroni: Instead of taking the unique, I take the new suit as opposed to 0 0 plus obviously, because it is always around
00:57:840Annalisa Cesaroni: of this measure all but the measure of the board
01:01:340Annalisa Cesaroni: less than Alpha, that actually, with a similar argument, one can prove that you have Alpha.
01:09:90Annalisa Cesaroni: Let's call it to be, maybe not a, because it
01:12:600Annalisa Cesaroni: see if there's something, it is different. Today.
01:16:470Annalisa Cesaroni: one can show this.
01:18:870Annalisa Cesaroni: Okay, one can show this. And so with a similar argument, the point is always using in an appropriate way for the theorem.
01:29:900Annalisa Cesaroni: Okay, this actually more or less
01:44:380Annalisa Cesaroni: in zoom of New of Bxr.
01:48:830Annalisa Cesaroni: Over the measure
01:50:990Annalisa Cesaroni: is equal to the name
01:56:180Annalisa Cesaroni: of new or the exact.
02:01:560Annalisa Cesaroni: and it is fine for almost every X. Why I have this. Why? Because actually
02:10:440Annalisa Cesaroni: once that we have property, one and property 2.
02:13:690Annalisa Cesaroni: Because actually, this is property, one and property 2
02:19:400Annalisa Cesaroni: by one plus 2. I have this. Why, this is 2, because I consider the set where this lim soup is infinite.
02:28:810Annalisa Cesaroni: Okay, 1st of all, I have to prove that this zoom soup is fine.
02:33:610Annalisa Cesaroni: Let's look at this where this is plus infinity.
02:38:680Annalisa Cesaroni: Okay, so actually, this. So
02:46:300Annalisa Cesaroni: is plus infinity.
02:51:130Annalisa Cesaroni: Let's call it C, then I have that nume of C,
02:55:560Annalisa Cesaroni: it's greater or equal, then also
02:58:00Annalisa Cesaroni: the back measure of C for every alpha positive.
03:01:580Annalisa Cesaroni: and and then
03:04:750Annalisa Cesaroni: and then this implies that, dividing by Alpha, that the Lebec measure of C is 0.
03:11:200Annalisa Cesaroni: Okay, we sent
03:13:468Annalisa Cesaroni: Alpha plus infinity. Okay.
03:18:360Annalisa Cesaroni: if I take the set of point where the Lipsoup is classifility.
03:27:620Annalisa Cesaroni: Oh, yeah, please. No.
03:31:920Annalisa Cesaroni: You think none of us.
03:33:770Annalisa Cesaroni: And then then I take the second point, where the premium is less than dream soup, and I use both property one and 2
03:44:450Annalisa Cesaroni: and see, and they take the set of point D, where, say, I, take a less than B
03:52:998Annalisa Cesaroni: limit, the set of Point X, where the limit are
03:58:830Annalisa Cesaroni: of new. Of these
04:03:230Annalisa Cesaroni: is less or equal than a less than B less or equal than billing soup.
04:13:10Annalisa Cesaroni: So for the measure of the bill.
04:18:10Annalisa Cesaroni: okay, take this set.
04:21:190Annalisa Cesaroni: and they have worked.
04:23:870Annalisa Cesaroni: I have that mu of D.
04:27:110Annalisa Cesaroni: Now, I just look at this, and I use property.
04:31:590Annalisa Cesaroni: One
04:32:870Annalisa Cesaroni: is less than important a
04:35:480Annalisa Cesaroni: times delivered measure.
04:38:30Annalisa Cesaroni: Okay. And from this I have that mu of T
04:42:540Annalisa Cesaroni: is greater or equal than B times. The leg measure
04:47:450Annalisa Cesaroni: they put together these greater than the the. So yeah.
04:54:220Annalisa Cesaroni: A
04:56:580Annalisa Cesaroni: sorry BD license A,
05:02:630Annalisa Cesaroni: but B is strictly bigger than A. So
05:07:490Annalisa Cesaroni: the measure of this has to be here.
05:10:470Annalisa Cesaroni: Okay, one
05:12:390Annalisa Cesaroni: act like this. And then the measurability of this limit. So this limit exists, defined almost everywhere.
05:21:10Annalisa Cesaroni: because the set of point where the limit is different from the is different from the length as measured 0. Moreover, I heard that almost everywhere the link soup is finite. Okay?
05:37:410Annalisa Cesaroni: And then. So
05:39:930Annalisa Cesaroni: now I have to do the disability. And I'm not going to do this because just for the it was just to give an idea what
05:50:980Annalisa Cesaroni: what is the way in which this kind of theorem and why we need to covering theories
05:57:450Annalisa Cesaroni: we need covering, because we need to use the measure so to use sigmativity of the measure say.
06:06:430Annalisa Cesaroni: okay. But as I was saying, this, theorem is not in the problem. Okay, theorem. The fact that this limit towards etc. is is affected as to be known, but the proof of his
06:22:570Annalisa Cesaroni: I'm not really
06:26:800Annalisa Cesaroni: okay.
06:29:100Annalisa Cesaroni: So the the things that have to be known is, say the coronary of the theorem of differentiation of measure. So the so-called vital bacterium. Okay, this is a theorem which has to be known.
06:47:500Annalisa Cesaroni: Okay? And So
06:52:980Annalisa Cesaroni: so maybe we are
06:56:160Annalisa Cesaroni: ready now
06:58:280Annalisa Cesaroni: to to define this so called out of measure.
07:05:300Annalisa Cesaroni: But before define the outward measure, I want to fix some notation, because to in order to define outward measure.
07:14:210Annalisa Cesaroni: I want to to define the automation by using
07:20:00Annalisa Cesaroni: renormization constant in order to define, to fix some notation. So let's fix some notation. Let's do an exercise based on the
07:32:500Annalisa Cesaroni: integration of sphere forward. Okay, it's just an exercise.
07:38:128Annalisa Cesaroni: So from now on, I report Omega, N, by definition. This is a definition.
07:44:710Annalisa Cesaroni: The manager, the back manager of the Board of Sense 0 one in a re.
07:55:540Annalisa Cesaroni: So Omega is a constant
07:59:80Annalisa Cesaroni: is the volume or say, volume
08:04:650Annalisa Cesaroni: of the board.
08:06:480Annalisa Cesaroni: Oh, God, this one
08:09:650Annalisa Cesaroni: in Arabic.
08:12:180Annalisa Cesaroni: Okay. And I want to write to write an explicit expression of this over again
08:19:910Annalisa Cesaroni: by using integration on
08:23:290Annalisa Cesaroni: fit.
08:24:450Annalisa Cesaroni: And
08:26:220Annalisa Cesaroni: so
08:29:399Annalisa Cesaroni: okay, he's a.
08:35:370Annalisa Cesaroni: So, for example,
08:39:189Annalisa Cesaroni: only the 2 is fine.
08:43:450Annalisa Cesaroni: The-the area of the
08:58:430Annalisa Cesaroni: 1st observation.
09:00:850Annalisa Cesaroni: 1st observation is the following one.
09:04:728Annalisa Cesaroni: if I consider the this, this sphere on our end, say, sn minus one in our end
09:15:202Annalisa Cesaroni: minus one, is what is just the point x 9
09:19:510Annalisa Cesaroni: with x equal to 1. 0, okay, I didn't say, but
09:24:160Annalisa Cesaroni: obviously the measure of the ball of center 0 around R is what is R to the power. M. Omega, M.
09:32:210Annalisa Cesaroni: This is due to the fact that the back measure is homogeneous of degree. And okay.
09:38:460Annalisa Cesaroni: So if I take the ball of radius instead of one r, it is like this, okay, on the left. So the measure of the ball of radius one at times.
09:50:600Annalisa Cesaroni: Okay, this is just by homogeneity. And and moreover, here, instead of 0, I can fix whatever center I want. Because, okay.
10:07:580Annalisa Cesaroni: so 1st of all.
10:10:400Annalisa Cesaroni: so this is a sphere. So the set of point with.
10:14:780Annalisa Cesaroni: So in our tool, it's just the simple no
10:19:80Annalisa Cesaroni: is just the simple, which is the relation between the 2 I have that of Sn minus one is just. M. Oh, my God!
10:30:320Annalisa Cesaroni: Why, this is true! This is true by simple application
10:35:180Annalisa Cesaroni: of what of the integration formula.
10:40:820Annalisa Cesaroni: Okay.
10:42:590Annalisa Cesaroni: integration of spheres forward. So I have a
10:49:440Annalisa Cesaroni: I have worked. I have that Omega N is lost is the measure of the ball of center 0. And I just one. So it is the integral over
10:59:80Annalisa Cesaroni: 0 1 Dx.
11:01:640Annalisa Cesaroni: so this is.
11:03:570Annalisa Cesaroni: if you want. This is the integral over Rn. Of the characteristic function of the ball. No. In X. Dx. This is a function which is regular
11:14:660Annalisa Cesaroni: for sure, so integration.
11:18:160Annalisa Cesaroni: Well, she have some.
11:24:870Annalisa Cesaroni: What does it says that says that this is equal to the integral between 0 and one of what they integrate over
11:32:290Annalisa Cesaroni: SN, minus one.
11:34:680Annalisa Cesaroni: yes, sometimes, and minus one.
11:39:820Annalisa Cesaroni: Okay.
11:41:670Annalisa Cesaroni: this is the the integration of
11:46:210Annalisa Cesaroni: and just taking the integral between. It should be the integral between 0 and plus infinity. But then I have the characteristic function of the ball centered 0. And so it is what it is. One over. N
12:01:853Annalisa Cesaroni: just integrated this No. One over N after the power S between one and 0. So one over N area of the sphere.
12:11:350Annalisa Cesaroni: because this integral is just the area of the screen
12:17:510Annalisa Cesaroni: aria is there.
12:19:600Annalisa Cesaroni: The area. If we add enough, 3 is just the area of the 2 dimensional set in our end is the area of
12:29:160Annalisa Cesaroni: yeah.
12:30:110Annalisa Cesaroni: Now, this is 1st 1st thing, then, actually, instead of writing the area of the sphere, I will write L. Omega. Then I want to find out.
12:43:350Annalisa Cesaroni: Ugh!
12:44:870Annalisa Cesaroni: A formula for this omega n, depending on the gamma from the Euler function that maybe you already have seen. But
12:53:990Annalisa Cesaroni: okay, so the the idea is this one.
12:58:50Annalisa Cesaroni: I consider this function. This, this function here. E to the minus x squared.
13:02:810Annalisa Cesaroni: Okay.
13:05:115Annalisa Cesaroni: x, 9.
13:08:690Annalisa Cesaroni: Okay. And then I consider the integral over Rn of X, of of this in the X.
13:16:160Annalisa Cesaroni: Okay.
13:17:885Annalisa Cesaroni: this is what
13:20:00Annalisa Cesaroni: this is what the
13:21:550Annalisa Cesaroni: I can apply, the integration of sphere formula. But it's a bit I will apply it after. But then this is what this is actually by.
13:37:470Annalisa Cesaroni: This is what e to the minus x squared can be written E to the minus x, 1 square
13:45:470Annalisa Cesaroni: e to the minus x 2 squared is the product E to the minus xn squared. Now, if x is x 1 xn, okay, so it is the product.
13:56:880Annalisa Cesaroni: and there are product of function.
14:00:00Annalisa Cesaroni: Each of these is equal, and it is depending on these.
14:04:970Annalisa Cesaroni: on this. So it is actually integral over r of E to the minus x, 1 square
14:14:430Annalisa Cesaroni: e to the minus x, 1 squared x, 1 to the power. N,
14:20:466Annalisa Cesaroni: it's just the integral of our app
14:24:410Annalisa Cesaroni: of the function E to the minus x, 1 square.
14:27:540Annalisa Cesaroni: the x 1
14:29:00Annalisa Cesaroni: T square. If you want in the T,
14:31:430Annalisa Cesaroni: what?
14:33:26Annalisa Cesaroni: 4? Okay.
14:34:940Annalisa Cesaroni: so now
14:36:620Annalisa Cesaroni: we want. We have this work, we have that actually, we have, we are able, we are not able to compute this integral, but we are able to compute Disintegra
14:47:290Annalisa Cesaroni: E to the minus x square in the x.
14:51:100Annalisa Cesaroni: Okay, you remember, this is the classical things that they can compute by using a regular coordinates.
15:07:730Annalisa Cesaroni: And so this is a 5, you know.
15:11:270Annalisa Cesaroni: Okay, this is an exercise of alpha. Push, 2 polar coordinates. This comes out to be pi.
15:18:220Annalisa Cesaroni: So actually, since
15:21:900Annalisa Cesaroni: the only thing square
15:24:150Annalisa Cesaroni: is equal to the integral over R
15:27:270Annalisa Cesaroni: E to the minus square of T
15:31:22Annalisa Cesaroni: to the square.
15:33:50Annalisa Cesaroni: And actually, this is square root of 5. Okay, square root of 5.
15:40:490Annalisa Cesaroni: So this is this is equal to pi to the power n, over 2
15:46:66Annalisa Cesaroni: ending up.
15:47:290Annalisa Cesaroni: because this is square root of 5.
15:49:880Annalisa Cesaroni: This is square root of 2 pi and to the power. N,
15:54:260Annalisa Cesaroni: okay, so this integral is just this.
15:58:350Annalisa Cesaroni: okay, so let's go today. A integration of sphere formula to the right disintegra. Okay.
16:07:491Annalisa Cesaroni: so let's rewrite this integral by using integration.
16:16:970Annalisa Cesaroni: So if you don't remember that this integral in up to is equal to pi.
16:21:890Annalisa Cesaroni: Okay, don't. That is just you have to rewrite.
16:26:890Annalisa Cesaroni: So you have to just revive by using the color coordinates and the angle. So you have 2 pi times. Okay, but if you don't remember, let's take a solution
16:46:570Annalisa Cesaroni: so
16:48:630Annalisa Cesaroni: integral over. So yeah, pi to the power. N, over 2 equal integral over M. Of E to the minus x squared in the X
16:56:930Annalisa Cesaroni: integration of fear
17:02:110Annalisa Cesaroni: is integral between 0 plus infinity R to the power n, minus one
17:07:100Annalisa Cesaroni: e to the minus r squared.
17:11:440Annalisa Cesaroni: It is a
17:15:300Annalisa Cesaroni: this is called sn minus one.
17:24:100Annalisa Cesaroni: Okay.
17:27:480Annalisa Cesaroni: and bizaria know.
17:30:880Annalisa Cesaroni: So this is the integration of sphere formula, no integral between 0 plus even Rn. Minus one, the function compute
17:39:146Annalisa Cesaroni: and defined in there
17:41:330Annalisa Cesaroni: the boundary of the sphere. But here the angle is not appearing, because the the function is symmetric times, this area. And this area is N, Omega, n.
17:52:940Annalisa Cesaroni: so yeah.
17:55:290Annalisa Cesaroni: 5 to the power n, over 2 equal to N. Omega, n, which is the
18:01:500Annalisa Cesaroni: is that area.
18:03:350Annalisa Cesaroni: Okay. Times the integral between 0 plus infinity of R to the power M minus one, the minus r squared in here.
18:12:764Annalisa Cesaroni: Now.
18:14:890Annalisa Cesaroni: this is the Euler function
18:17:920Annalisa Cesaroni: computed that at N over 2 y. Times 1 1,
18:22:990Annalisa Cesaroni: I have to change the variable. Let's change the variable. So let's take s equal to r squared.
18:31:878Annalisa Cesaroni: So oh, my God.
18:36:310Annalisa Cesaroni: t to the minus s
18:41:950Annalisa Cesaroni: s.
18:44:688Annalisa Cesaroni: Minus one over 2
18:46:660Annalisa Cesaroni: s.
18:48:918Annalisa Cesaroni: S. Plus one alpha s. Minus
18:54:960Annalisa Cesaroni: s. Minus one off
18:59:225Annalisa Cesaroni: should be what I've done.
19:04:60Annalisa Cesaroni: Okay.
19:08:464Annalisa Cesaroni: change the variable.
19:10:50Annalisa Cesaroni: And then and then actually.
19:16:330Annalisa Cesaroni: what is this? It is n omega n. Over 2. I take this to outside, integral to 0, plus infinity of E to the minus S.
19:27:470Annalisa Cesaroni: Thanks.
19:29:360Annalisa Cesaroni: S.
19:31:222Annalisa Cesaroni: And what to and
19:34:740Annalisa Cesaroni: over to minus one.
19:37:920Annalisa Cesaroni: Yes, still
19:40:798Annalisa Cesaroni: and over 2 minus one.
19:47:00Annalisa Cesaroni: I did that correctly, I think. Yes, no.
19:52:625Annalisa Cesaroni: Is- is one.
19:56:675Annalisa Cesaroni: So
19:58:460Annalisa Cesaroni: yes, no.
20:00:400Annalisa Cesaroni: And this is what
20:04:930Annalisa Cesaroni: this is
20:08:830Annalisa Cesaroni: the Euler function. Okay.
20:12:820Annalisa Cesaroni: so it is.
20:14:450Annalisa Cesaroni: Omega n. Over 2 gamma of N over 2.
20:20:230Annalisa Cesaroni: And it is actually omega times gamma l. Over 2 plus one.
20:26:550Annalisa Cesaroni: What is the Euler function. The Euler function is this function defining delta plus infinity can also be extended to the complex case which interpolates the the
20:39:480Annalisa Cesaroni: which interpolates the the
20:41:890Annalisa Cesaroni: okay.
20:42:820Annalisa Cesaroni: The factorial. No
20:52:50Annalisa Cesaroni: gamma of X is called Euler function.
21:00:830Annalisa Cesaroni: Gamma function. Sorry Gamma function.
21:07:170Annalisa Cesaroni: It was defined by Euler
21:11:120Annalisa Cesaroni: by the mathematician oil
21:13:610Annalisa Cesaroni: in the seventies.
21:15:990Annalisa Cesaroni: and it is integral between 0 and plus infinity. E to the minus s.
21:22:310Annalisa Cesaroni: s, to X minus one. Yeah, yes, it's okay.
21:29:211Annalisa Cesaroni: is equal to gamma of x plus one.
21:43:270Annalisa Cesaroni: So actually on gamma of N is equal to N minus one.
21:50:179Annalisa Cesaroni: So again.
21:51:650Annalisa Cesaroni: okay, the mobile.
21:54:280Annalisa Cesaroni: So actually, we end up
21:57:380Annalisa Cesaroni: observing that Omega N is actually pi to the power. N. Over 2 over gamma.
22:05:180Annalisa Cesaroni: Oh, N. Over 2 plus one.
22:09:504Annalisa Cesaroni: Okay.
22:11:410Annalisa Cesaroni: this will be useful in order to define the-the-box definition of observations.
22:19:460Annalisa Cesaroni: So if I'm going to facilitate, it is good to 0. Okay.
22:26:940Annalisa Cesaroni: okay? So maybe we have no more time.
22:29:920Annalisa Cesaroni: and tomorrow I will start with the out of measures which are measured, which are defined.
22:38:30Annalisa Cesaroni: which
22:39:590Annalisa Cesaroni: are measure defined in all the subsets of R over n, and which are useful, which will be not in general problem measure, and which are useful to measure sets in our N with the dimension less than n.
22:58:170Annalisa Cesaroni: okay.
23:00:190Annalisa Cesaroni: okay, so this is, this was, maybe a lot of you have already seen this computation was just an exercise. Okay, so let's
23:10:40Annalisa Cesaroni: we will continue tomorrow with the thousands.