 I want to add, just as the other lecturers have said, a very warm welcome to Trieste, and over the course of the two weeks I very much look forward to interacting with all of you personally. This is lecture three. So, it's introduction to our Gothic theory. We have five lectures as this course, but each of us are only doing one lecture each. So, I wrote our names up here. The order is chronological. I'm talking today. So, well, we all stood up earlier, but maybe I can ask my fellow lecturers and tutors to stand up again. Llucia, arena. So, yes, it's in order. So, today, me, tomorrow, arena on Monday, on Tuesday, and then, ah, Stefano is joining us. And Llucia at the back. And Davide on Friday. And we'll also be the tutors during the exercise classes. And all of us very much believe you can't really listen to mathematics and you can't just read it. Mathematics is about doing it. So, well, you guys are very calm and quiet. And I think you're paying attention or sleeping. And it's very respectful. You're being very quiet. I came in here to try and work out putting the PDF on the computer. And it was really calm, really quiet. I felt maybe it was very, very respectful. So, before mathematics, I have a task for all of us. So, it goes like this. In just a moment, talk to the person behind you, the person to your side, the person to the other side, the person in front. Introduce yourself to these people. And one additional rule. Find out one piece of information at least from these people beside you in all directions. It can be maybe where they come from. Or perhaps more important details. Whether they're preferring pasta or they're preferring pizza. Or perhaps whether in their home country the summer is hotter or colder than here in Trieste. Rules are clear? OK. Start. What should we do? Everyone to the left. Everyone to the right. Go, go. In all orders. Simon in front. Good, good. So, now I apologise for interrupting you. But yes, this is good. You're going to have fun without the mathematics. You'll have far more fun without me saying anything. I have to stop you. I have to stop you because otherwise everyone else will think I haven't prepared anything to say. And I'm just trying to find ways to escape. I have a lot of lectures I haven't done very well, but... So, this is... So, our course is Introduction to Agotic Theory. And to start with, we're going to do measure theory today. I'm not really going to do much about dynamical systems at all, which... Yes, it has to be the way. And I'm not really going to motivate it so much either. I'm going to try very briefly, but we've had the two lectures this morning. Maybe we're getting some idea. I wanted to say very quickly and very loosely, and therefore I want to remove the slide quickly before anyone points out problems, that as we've seen earlier and as many of you have seen before, the framework is we're looking at dynamical systems. Questions like maybe some orbits we can understand, but maybe then other orbits they're not quite the same. And then what about typical orbits? And definitely I needed quotation marks here or many quotation marks. We want to say something, so maybe a measure is going to help us with this aim. And also, I've written here the push forward. So, if we have a measure and we have... Well, it has to be a measurable map, then we can look at the push forward actually on the space of measures. And so we can imagine maybe we're faced with a very complicated dynamical system, but then actually we have a linear map when we're looking at on measures. So as is the sad truth of all these problems, if we make one problem disappear and it became a linear map, that's lovely for sure the other problems. There's here somewhere, they just moved to a different place. But yeah, I can't say much. We're going straight into measure theory. We're going to believe it's important. I'm sure all of you here have seen measure theory before in one way or another. Be patient, be tolerant. We need it set up so we're all talking the same language in the same way for the rest of the lectures. The goal. It's not really a goal. But I wanted to start somewhere. And as I'm not starting with dynamical systems, maybe measure theory a goal. So it's not really a goal. It's sort of a vague idea. But let's start somewhere. And one of measure, measure the etymology or something like this. We have the idea of volume or length for things like this. We want to formalise this properly as maybe an intuition. Yes, lots of intuition, which means incredibly dangerous from a mathematical point of view. When there's more intuition present, it's more difficult. It's something like this. If we're looking towards Lebesgue measure, which we're only doing for the moment, we want it so on RN. I'm thinking on RN, Euclidean space. We're looking for something that conforms to what we know on the sets we know about. So for instance, the unit cube, it should have unit volume. We want to have the idea that stuff matches together. If we have sets and other sets, we want to take the unions of them. We want to be able to measure them separately and add it together. If I take the measure of one set, like the volume, and I look at a subset, then I want the measure of the subset to be definitely not bigger than the measure of the big set. Things like this. And also in the case of volume in Euclidean space, we want translation invariance. But where do we go with this? In actual fact, we would like to do this for every subset of RN. It's not actually possible if we want all of this. What's the best that can be done? And then what happens if we drop some of the requirements I've listed here? Maybe we don't care about translation invariance. Maybe we don't care about some of the other things. And this is measure theory. So we get serious. Actually, maybe. Who knows what an algebra is? Who knows what a sigma algebra is? So anyone with a hand up, I can ask him quickly. No, we go there anyway, but fairly fast. So we're looking at subsets of our space. We want the empty set to be one of them. We want it to be an algebra in the sense that if we take compliments, the compliment is also in the set. We want it to be closed under finite unions like this. Oh, so simple quick question. Someone say very quickly, if I want to change this to a sigma algebra, what on the board am I changing? Countable. Indeed, sigma algebra, then we do it for countable unions. We also need for talking in a moment that if we take some collection of subsets, then we write this notation. We can take some collection and then we can say, well, it's not big enough to satisfy what we had there. We add some more. So we will call this the sigma algebra of that collection of subsets. So definitions are fine, but really they're just decoration around examples. Many examples. So on the line. So we take the collection of all finite unions of subintervals. It's an algebra? Is it an algebra? Yes. No? Yes? Yes. Is it a sigma algebra? What about we again take the real line and we look at the collection of all the subsets of the real line? Is this an algebra? Sigma algebra? Very simple examples. Now something just a little bit different that we use all the time. The terminology Borel sigma algebra, how many of you is known to everyone most? For the Borel sigma algebra, for any topological space you have the sigma algebra, the smallest sigma algebra that contains all the open subsets of that space. Good. We're covering a lot of ground. So yeah, very basic things just to be at a good stage on the terminology. We have a measurable space. A measurable space is the space and the sigma algebra of subsets. A measure. So a measure is this function from the sigma algebra to the extended real line. We want to know that the empty set has measure zero and we also want this, we call it a countable additivity or good or clear. That's our measure. Well, before I show anything, examples of measures, favourite examples of measures. Let's shout out some examples of measures. Lebag, something else? Zero measure. I like it. Elegant. Yes, the delta measure typically cool. It seems reasonable any Greek letter. Oh, I forgot, I was going to say. Yes, for a later reference, we differentiate, I said to the extended real line. So we differentiate between, well, let's go on. So this is what you said. You came before. We come back in a second. So I wanted to say we have finite measures. What's an example of a measure and on a measurable space that's not finite? We keep using the real line. So maybe a measure on the real line. Lebag measure on the real line. Simply because the real lines, it's big, it's long. Probability measures. Actually, when we are talking about dynamical systems, we're kind of seeing all the time a connection to probability theory in some way. So the word popping up is not surprising. It's a terminology that if the measure of the whole space is one, it's a probability measure. Of course, if it's finite, we can renormalize it. And in the definition of the measure, renormalizing it still would be measure. So, yeah, examples, you mentioned the direct delta. At the very beginning, I listed some, I said goal. I said maybe it's a goal. I said this is the sort of thing one was aiming at when one was trying to create Lebag. But then we're seeing where it goes anyway. In what way does this fail to satisfy what we were thinking about at the beginning? The translation invariance. So, yeah, maybe it's good for things, and then it's maybe not good for other things, but not translation invariant. Yeah, I wanted a nice way for this to crop up, but I wasn't sure how to bring it in in a neat way. So we, and we haven't seen enough examples. Okay, it's coming in the wrong order. But we've understood that for some reason we're looking at space and we're taking a certain collection of subsets. I haven't really, I realize I should have convinced you why we're not taking all of them. I told you we can't take all of them in some cases for some motives. And then I just started not taking all of them, talking about a collection which isn't everything possible. Hopefully it becomes clear when we really play with it. So here a useful concept that maybe there's a measurable set. So the terminology is any subset that is in the sigma algebra is a measurable set. So here maybe I could have a measurable space, a measure space, and I have a measurable set that has zero measure, and then I want to look at a subset of it. And maybe some of the subsets of that zero measure set aren't even measurable. So for sure if it was measurable it must be measure zero, but maybe it's not even measurable. So yet another question. Can someone give me an example of a non-complete measure space? Are people just really quietly replying, or is this definitely a question slightly further forward? Any ideas, possibilities? Ah, so just add to your saying just to check if I've understood correctly. So we have a measure space and we say, oh no, it's not complete. Let's just add all the zero measure subsets. Can we definitely do it? Is it still going to be a sigma algebra when we add all of that? So yes, we want to do this. I don't have a really proper concise answer, and I'm sorry about that. There are various extension theorems that can. So yeah, I'm sorry, I don't have a really concise. It is something to worry about. Yes, in some way we want to have all the zero measure, all the subsets present in the sigma algebra, and we want to assign zero measure to them. I think there are some difficulties knowing that you can do it just in general. But I believe there are extension theorems that I don't remember precisely. Any answers from the audience? Ah yes, for the example, Lebesgue measure, and the Lebesgue measurable sets and the Borrell measurable sets. So yes, Lebesgue measure on the Borrell measurable sets is not complete. Yes, I try and answer you more fully another time. I would have to think, in general it's not so easy. Okay, going on. So I wanted to say, because we're always dancing around Lebesgue, I wanted to recall a construction of Lebesgue measure. There's various ways. Well, not so many ways. Who for Lebesgue measure prefers to define outer measure in inner measure, and then define Lebesgue measure, and who prefers to define outer measure, and then use the Cattitidori system. First, outer and inner. Second, just outer and then Cattidori. Good, good. Okay, that's the way I was going for. Thank you for trying to make the system better. We still have sides and sides. No, okay, it's better. So, yes, I'm looking just on R now, because from the first two lectures, I realized we are in a one-dimensional week. Although everything I'm saying applies to a higher dimension. It goes through basically the same. So yeah, outer measure defined like this. So we look at sets of intervals, which cover the set we want to measure, and we take the infimum of those possible. So we understand for an interval, we can really sensibly define the right measure of that interval. It's just the length of the interval. And so we define outer measure. Is outer measure a measure? Not according to what I said before. It has some properties. I'm not going to look at it now. I'm going to say you've probably already checked this in your past. It's sub-additive. We were talking about additivity before. In this case, outer measure is sub-additive, and note well we're defining it for all subsets. So then to construct a big measure, we do that, and we check the properties of this outer called measure, but not actually outer measure. And then we do this. We look at all the subsets of A such that this holds. So does it seem reasonable that this holds? Let's think. So if it was a measure, yeah, so what we're doing, we're looking at the intersection, and we're trying to measure a set A. What are we doing? We're trying to measure a set A. Then we're looking at some other set E. Maybe it just covers the whole of it. What happens there? Okay, E covers the whole thing. Ah, so the intersection is the whole of A. And what about this other bit on the right? Sorry about the star. The star obviously fell down. It's meant to be up the top. So that other bit. Complete E without A. Ah, but that's just E. Okay. Have I got this right? I've made a mistake. This should be the other way round. Yes? Does that seem reasonable? This was an intentional exercise to check whether... I'm sorry. Oh, I want E there. I'm sure I want E there. I'm sorry. Is it good? I want us to be happy it's right. I'm sorry. Yes. So it seems reasonable. Do you think it contains a lot of the subsets? It turns out it contains a lot of the subsets of R. So in the construction of Lebesgue, what we do is that. So that is the sets that are Lebesgue measurable. I then take the set of all Lebesgue measurable sets and, well, I'm going to hope and I should prove it's actually a sigma algebra. So yes, here we have a sigma algebra of all the Lebesgue measurable sets. Of course it has to be checked. And then for these sets, we define the Lebesgue measure to be equal to this outer measure of that set. So yes, we said that exercises are the important part of the week of doing mathematics. So here's... We have two exercises and each exercise has two parts. So this is the first of the two exercises, exercise A. And this is something a bit different. So here's an example. I don't know whether this is a very common example, but I saw it recently and it seemed interesting to me. So this is the example. Is it clear? So we have the set of natural numbers, is my space, and then I look at all the subsets of these natural numbers where either that set is finite or the complement of that set is finite. Then I define this function mu. So of course I'm using a symbol that I like to use for measures. So I'm thinking maybe this is a measure. But anyway, that's what it's defined as. And so the question here is, is this function additive and is it countably additive? Countably, sorry. The exercise is clear. The other thing that I said had to be shown and I didn't suggest anything about how we're going to do it. I did mention about the sub-additivity of outer measure. So yes, this is a second part of the exercise. Again, we're talking about algebras, sigmar algebras, show that the collection of Lebesgue measurable sets is a sigmar algebra. Or go with the exercise. So yeah, when it comes to the afternoon, I think we will probably write the exercises on the board. We will have, yes, we have from each of the lectures today, we will have two exercises from each lecture. And when it comes, we have all the same time for all of them. And in theory there's going to be one slightly easier and one slightly harder exercise from each of the lectures. But of course this is going to be subjective. We see what happens this afternoon. So yes, exercise. Good. Another thing that's very important for us to, well, I haven't really, there's something a bit more I want to communicate that, so we're fighting. We, with the business of Lebesgue measure. And now let's recap for a second. So we're looking at Rn. But actually I'm saying Rn is like R. We look for today just at R. And on R we have Lebesgue measure. We've always used Lebesgue measure. We can happily live our entire lives thinking about volume. As physicists we can happily live our entire lives believing that we're rigorous and thinking of Lebesgue measure without really questioning too much. And here we are thinking how actually we can't say all of the subsets of Rn are measurable. We can just say most of it for lots or quite a lot or enough for our purposes today. And we've said about the, so let's clarify the two things. We have the Borel sigma algebra and we have the Lebesgue sigma algebra. The Lebesgue sigma algebra, the sigma algebra of the Lebesgue measurable sets. And these are not the same. Well, maybe that's an exercise. Yeah, well, let's see. So it's not quite clear what, because it's not obvious in any way, but if we're making so much effort about it there must be some sort of craziness going on. And these sets are looking a bit weird. They're looking a bit counter-intuitive. So, we don't want to study measure theory this week. We're doing dynamical systems. What we want to do is be able to show results when we're working with measures. So this is one of the main motives for this. So this is a theorem. So the theorem, we start with this setting. We have a mu star, a function which satisfies this. Yeah. So I've insisted that it's finite, but I see in the top line I've ensured it's finite, because I've ensured it's a probability measure. Well, at least not. Yeah. So it is some function that assigns to the empty set, the value zero. As I say, it's definitely finite because I wrote that in the above line. Then if I take countable collections of pairwise disjoint measurable sets and if the countable union is a measurable set. So, of course, I've said algebra at the top, not sigma algebra. So it's entirely possible this countable collection, the union might not be a measurable set. So this is just in the cases where it is a measurable set. So if it is a measurable set, I want this countable additivity to hold. If that does, then the theorem tells me that there exists a unique measure, and, of course, sorry, that I was going to do it in one phrasing then in another phrasing, so this is why it's been mixed. Of course, this other measure is from zero to one. It's a finite. Anyway, there exists this unique measure on the sigma algebra generated by the algebra which extends the measure we start, the, so, pre-measure. The terminology is pre-measure. So if I have these properties, then I can get the extension, and it's unique. So it's good to have this because quite soon we'll be checking, oh, we've got this system and it goes this and it push, maybe push forwards this measure here, and so we have a measure defined in one way and a measure defined we started with, and we want to say, are these two the same? If we have to check it on a lot of measurable sets, then this might be a bit hard. This says we only have to check it on a limited amount, and we know because of the uniqueness that suffices. Please, Milify. I'm sorry. Ah, no, I'm sorry. This is stated for general measures, not just for the outer measure defined before. Yeah, sorry. Yep, this is starting from nothing. This is any general function, mu star. It happens to have the same symbol as used previously. Thanks for pointing that out. So, another set of exercises. So first of all, very simple. Show that there are subsets of R which are not Lebesgue measurable. And I want to do it. Hang on, I have the other one, but then I pop up a hint. Yes, the hints are coming after. So, my suggested way to do it is consider an irrational circle rotation. So, the one that Karina wrote here first thing this morning. So, we want to use the irrational circle rotation. And we have orbits. She talked about orbits. Some orbits happen to coincide and some are distinct. I am choosing a single point on each distinct orbit and from a single point on each... Yeah, let's see. Is it clear how to do the whole problem now? I hope not totally. No, there are some details to look at. So, I hope it's an enjoyable exercise. And we're also getting the dynamical systems idea back into the measure theory which we're doing in order to do for dynamical systems. It has a certain rightness about it. Number two of the second exercise. Show that there are Lebesgue measurable sets which are not Borrel measurable. Because why not? Whilst we're at it, let's play with it and do everything. Do we want a hint? No hint? I'm sorry. Oh, maybe I didn't. Sorry, thank you for the question. So, the question was about the definition of the Borrel measurable sets. So, this is you look at for a topological space, you look at the sigma algebra generated from all the open sets. This is the Borrel sigma algebra. So, a Borrel measurable set is one of the elements of the Borrel sigma algebra. Ah, the Borrel measure will be Lebesgue measure. Ooh, this points out... Yes, it's been in the background and I haven't said it explicitly. So, of course the definition of the Borrel sigma algebra can apply to any metric space. Let's, for the purposes of today, fix what we're talking about. Sorry, metric space, any topological space in particular metric spaces. But let's go back a bit. Let's talk about R. So, the spaces are or a subset of R or we can see the circle as a subset of R. So, here we've already discussed about Lebesgue measure. And we have... Yeah, I'm very happy you asked this question because it really makes me say the thing I really should have said without anyone pointing out. So, what are we going to do? So, we've got here Lebesgue measure, we discussed outer measure, we discussed this criteria for the definition of a Lebesgue measurable set. Now, let's go and start back from the beginning. So, I have R. Now, I look at the space of the set of the collection of all the open subsets. Now, I look at the Borrel sigma algebra which is generated by this collection of open subsets. You know I could do something because, well, let's suppose I looked at intervals and the intervals will generate the open subsets for me. So, let's even take a step back. So, here what we're going to do is we have intervals. We can very easily define the measure of intervals by length. And we can very easily define... Ah, yeah, but there was the example the collection of all the finite unions of subintervals. So, we could define a measure on this quite well because we have a really proper notion of length of the subintervals. We can certainly add them up. So, yes. We have now, what have we got? We've got this algebra which is the collection of all the finite unions of subintervals of R. And we've defined a function on this using length in a very straightforward way. This, it's not so hard to check, has the additivity property that we've talked about a lot. So, it has an additivity problem. And we have this algebra. And actually you can see this behaves properly even if we take countable unions of elements of this algebra if that union is still an element of the algebra then this function we've defined this coincides like we required. So, now we can look at the catatio dori extension. We can look back at this. So, we're now applying this. We're applying it to the algebra which is the union the collection of all finite unions of subintervals of R. And we define this mu star in this case based on the length of intervals. Now, we're checking that this second point holds. And we can show it does hold. So, now we can see the extension which holds on the sigma algebra. So, in this case that sigma algebra will be the Borel sigma algebra. So, we could define this measure in this way. It's a measure which coincides with length on all the intervals and it's a measure to find on the full sigma algebra. In actual fact this measure also coincides with Lebesgue measure. So, it's not absolutely clear but we have here the situation that the so, in this setting where we're looking at are the Borel sigma algebra will be what's the right word? Not subset. The Borel sigma algebra is strictly smaller than the Lebesgue sigma algebra. But we can look at the two sigma algebras with the same measure. So, in both cases it's Lebesgue measure. I hope that. But thank you for bringing that up. It's something I should have said before. So, what does I... Ah yes, we exercise we didn't want any hints for the second part of exercise B or we wanted hints. So, I don't know the hints are quite short. I don't know how short or long it is. We'll see and I will definitely be present during the exercise classes. So, I guess if I give exercises that just really horrible I suffer maybe physically later. Anyway, I'll be here when we do all these exercises. So yes, my hints are simple. We will call the Cantor function to consider some pre-image. I don't know exactly what's coming in the next days for the dynamic systems thing. I think we're also going to do integration. I... What's going to say a bit about integration. We do like this because it's coming to lunch time and in theory I have two minutes left and it's okay overrunning a lecture a bit but overrunning a lecture before lunch and anyway I need lunch as much as you guys want. So, at some point we will be looking at integration and that's the other little bit of measure theory I'm not covering here in any way. But yes, so be it. At some point we play this. We define integration. We talk about characteristic functions. We then talk about simple functions and then we would define integration for positive functions. Then we have a concept of integration. Just in the moment we don't have time to do justice to this and we can also talk if it's required. Okay, so yes for the moment if anyone has any questions about anything let's please ask and yes, you're good. Why did we make so much effort? That was meant to be where we came back because we found the problem then we worked hard and then why did we make so much effort with Lebesgue? Lebesgue is complete. It would be very good. Yes, so it's a good exercise but I never said it. Lebesgue measure is complete. I've not presented an argument for it. That's why we spend a lot of effort with it. In my point of view that's the huge thing. The sigma algebra is strictly bigger than the Borel sigma algebra and it is complete and yes, he's very right with the observation in the second part. We want to use that because what if we can find some subset of a zero measure set which is not measurable? Any other comments, questions? Lunchtime. Thanks very much.