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This is the prelims course called Analysis three. And this course is about integration.

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So let me start by telling you a few things about what's going to be in the course.

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So essentially, this course is about making rigorous the notion of an integral.

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So how can we make the notion of an integral, rigorous cost control?

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And so basically,

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what it was I'm going to be doing is showing you how to make rigorous sense of lots of facts that I'm sure you've all known since school.

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So we will make rigorous sense.

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Of facts such as. Well, things like that, the integral of X squared from zero to one is one third and even what that means and then things,

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I'm sure you all aware of that the basic fact that integration and differentiation opposite operations,

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so integration and differentiation are opposites.

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So we'll formulate that really carefully and prove a couple of versions of it.

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And then we'll also look at some basic functions, which are best defined using integration.

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So we'll look at the exponential and logarithm functions.

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And then we'll finish by looking at some things that you've not only known since secondary school,

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but since primary school, such as the fact that the circumference of a circle is two PI.

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So the circumference of the circle of Radius one is two PI and that the area of a circle of Radius one is PI.

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So actually today to even define what the circumference or the area of a shape is,

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unless it's a sort of rectangle or something like that, you really need to know what an integral is and actually what's PI?

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So we'll also be carefully defining PI and showing that it's equal to these two things.

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When those two things are carefully defined. So these are some of the things that are going to be in the course of time.

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So another comment I want to make. It's a bit of an apology, really.

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I'm apologising not for myself yet, but for the subjects of mathematics, really,

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so the integral that we're going to talk about in this course is not the right integral.

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So we'll be talking about. What's called the Raymond Integral?

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And it has the advantage that it's quite intuitive.

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It's relatively intuitive. And also relatively easy to define, so relatively, but it has some shortcomings, which we'll also be discussing.

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So it has various shortcomings, not the least of which is that there are some functions that you'd like to be able to integrate, but you can't.

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And as a result of that, I think it's fair to say that professionals would always use what's called the LeBec integral.

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So professionals, whoever they are, tend to use the LeBec in school.

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But then the package goes quite a bit more difficult to define. So what why are we spending eight hours talking about this second rate integral?

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Well, I think I've already mentioned it's relatively intuitive and easy to define,

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but thankfully whenever this remittance bill exists, it's equal to the baggage.

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So we are actually talking about a subset of the the kind of full theory that professional mathematicians would use.

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So this is a little bit like having to learn to crawl before you can learn to walk.

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Now, an apology that's more of a personal apology, but it's not really an apology, just a comment.

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I'm actually not going to talk about the remaining screw. I'm actually going to talk about something called the taboo in school.

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And the reason I'm going to do that is that I find this the the best way to develop the theory.

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I find it just the most intuitive way to talk about the theory.

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And it's then a theorem that will prove later in the course that the top goal is the same as the remaining school.

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So this turns out to be the same. As the minutes grow, but that's a theorem that has to be proven.

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So one consequence of this feature that I'm going to talk about something that's not exactly the same as three minutes grill to begin with,

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I would urge you to use the utmost caution if you look at any books about this subject early in the course,

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so you don't really need to because there are four lecture notes for the course on the internet.

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But different books take subtly different approaches, and it will get very confusing,

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at least to begin with if you try and compare what I'm talking about with some books.

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What would be good is at the end of the course to have a look at a couple of books and see different ways of developing the theory that,

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OK, so I think that's what I would say by way of introduction. So let's make a start on some actual mathematics.

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So the first chapter is about step functions and the integral.

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So as I'm sure you all aware, the integral, whatever it is of a function,

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is supposed to be kind of measuring the area under the graph of that function, whatever that means.

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And so what I'm going to do to begin with is look at some extremely basic functions and write down what they're integral surely would have

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to be for any vaguely sensible definition of integral so that these functions that I'm going to look at are called the step functions.

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So here is the definition, a function fi from a b to the rails, so I'll only be talking about real value functions in this course,

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but one could talk about complex value functions without too much more difficulty. So this is called a step function.

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If there are some intermediate points between A and B. So a sequence of points, the first of which is and the last of which is B.

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I'm such that fight is constant on the open interval between X Y and Z plus one.

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For between zero and minus one. So that's all it is.

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I'll give a simple example at the moment, but just a quick note I haven't bothered to say anything about the value of fight at the Points X exi.

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So we don't care. What value?

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Fire takes at the south points outside.

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So just to make absolutely clear that we know what we're talking about.

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Let me give an example if I take, let's say, eight equals zero and vehicles two.

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So the function fly from zero up to two to the rear is defined as follows.

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So let's make it 10, it X equals zero. I don't know, three, if X lies between zero and one minus seven at X equals zero.

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Sorry, X equals one. I'm one.

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If Pax lives between one and two and then minus six, if X equals two.

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So that's a step function and we could try and draw the graph of that just to really belabour the point.

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So here is supposed to be some axes that zero this one, that's two.

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And then this function is going to look a bit like this, so it will be 10 zero.

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Three, on this open interval, there are minus seven down there.

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One here. And then minus six down here.

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So that's the graph of that step function, so that's what they all look like.

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We call a sequence of points that splits the interval AB into finally many parts that's called a partition.

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So a sequence of points. A equals x nought, that's three x one B is called a partition.

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Kelly P of the INS for Abby.

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And so we say that the step function is adapted to this particular politician.

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OK, so hopefully everybody is pretty comfortable with this definition,

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I've gone over it quite slowly because it's going to be absolutely fundamental in the course.

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We'll just be talking about step functions over and over again because they're what's used in defining the integral.

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And just before stating a simple lemma, I want to wait one more definition, which is a very natural one.

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And this is the notion of a politician being a refinement of another politician.

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And that's simply that you if you start with a politician such as zero one two here,

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then a refinement of it comes by just putting a few more points exi and keeping the original ones.

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So definition a politician primed.

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So that's going to consist of points Exide primed up to X and prime primed, and that's a refinement of P.

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If every X prise is an X, so if every X I primed is an X J.

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So that's the notion of refinement. It says clock on console is wrong.

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Is there a clock that's actually right anywhere? Does anyone have an opinion on what the time is right now?

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The 10 19. OK, so I'm going to make the assumption that this clock is wrong, but sort of consistently wrong.

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That's 17 minutes fast. So I'm going to stop the lecture at about 11 12, according to this clock.

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OK, let's record. A simple lemma about politicians.

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But his records and that. OK, so this is just very simple, basic facts, really, just to check that we've understood what the definitions are.

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So suppose that fi is a step function adapted to some politician p then if I take a refinement of p p primes.

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Then fire is also a step function adapted to be primed.

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Second, if I've got two different politicians, then there's a common refinement of both of them.

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So if I've got two politicians of the IDs for Abby, then there's another politician that refines them both.

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Common refinement. Pay.

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And then finally, just some. Closure properties of the space of step functions.

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If five, one and five two step functions.

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Then so are. Five, one plus five to scale and multiples, and then also things like the maximum nine five one five two.

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So this is this basically nothing to prove here? I'll make a remark about point three.

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And I'll talk about the other ones. So I think point one, it's basically obvious it's the case of recording what the definitions are.

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Point two is also basically obvious. You just take the points defining the position p one and the points defining the partition

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p two and throw them all in together to get your new set of partitioning points.

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And then three does deserve a little bit of a remark because these step functions may well be adapted to different politicians.

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So I'm going to just say that one and two are pretty obvious. And then three?

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Well, I should at least say which politicians are going on here, so I suppose that PHI is adapted to PI.

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I equals one two.

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Well, then by passing two a common refinement of P1 and P2, I can assume that five one and five two are relative to the same partition.

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So by passing to a common refinement. P.

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I can assume.

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That P1 equals P2 equals pain, and then point three is obvious because it's basically the fact that if you've got two constant functions,

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then there some is constantly scale multiple is constant in the max and the constant.

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So then this is obvious. Just looking at each of the sub intervals in the partition.

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OK, so there's a basic lemma, but the point of which is to say that this is a sort of a reasonable class of functions,

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at the least, it is closed on to some of the basic operations on functions. So the next lemma or just before taking the next lemma make a definition.

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If I've got a Set X on the rail line, then it's indicator function is just the function that takes the value one on X and zero elsewhere.

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It's indicator function. One Sub X is the function of taking values.

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Value one for X and X and zero.

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If X is not an X. That's quite a standard definition.

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And then the Lima number one point two is just the statement that the step functions

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are the same thing as the vector space spanned by indicator functions of intervals.

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So the space of step functions on a B equals the space of linear combinations, finite linear combinations, of course.

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Of indicator functions of intervals.

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So One Direction, I think, is really obvious, and that that's the fact that if you've got the indicator function of an interval is a step function,

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I think that is obvious from the definition of step function.

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So the indicator function of an interval. This is a step function, and I think that's obvious.

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And therefore, by the last lemma, a linear combination of indicator functions of intervals is also a step function.

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Hence, by Lemma 1.1, the third part, so is any linear combination.

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So that's one direction, and then for the other direction,

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I need to know that a step function is a linear combination of indicator functions of intervals.

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So conversely, a step function is adapted to a particular politician.

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So P. Well, it's by definition it is constant on the open intervals, excite up to excite plus one.

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And so it's a linear combination. Of while the indicator functions of the open intervals.

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And. The trivial intervals, which are just the points, exi.

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So notice that I was quite careful not to say whether my intervals were open or closed or half open or half closed,

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and so it's important here that I've allowed both open intervals and these closed intervals that consists of just one point.

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So let's just make that completely clear. Open interval.

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And this is a rather trivial closed interval.

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So that's a proof of dilemma.

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So occasionally, this space of step functions, which I've not described in two different ways, just straight the basic definition.

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And as the linear span of indicator functions of intervals sometimes but not that often given the name.

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So sometimes we write our sub step of AB for the facts space of step functions.

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On Abe. So let me test my clock hypothesis is the time now 10 30.

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Good. So general comments about this course is one of these bits of maths where.

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Once one has to go through a lot of very slightly tedious, really and quite simple lemon.

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But by the time you've gone through them all, there are so many of them that what you've ended up with is actually not that simple.

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So it's one of these things where locally very little happens, but globally. After a few lectures will have ended up with quite a nice theory.

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So that section was all about just what a step function is.

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And now I want to tell you what the integral of a step function should be, but I'm not going to phrase it like that to begin with.

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And so this section is called I of a step function.

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So just to make it clear that I'm not using some weird grammar, I mean, I as a function so associated to any step function,

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I'm going to assign a quantity called I, which I secretly know is going to be the integral,

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but I'm not going to allow myself to call it that just yet. So let me define what I'm talking about.

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So suppose I have a step function, so let five be a step function.

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Adapt it to. A politician, P.

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Between the on on a up to B and let us suppose so, by definition, it's constant between X, Y and Z.

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Plus one. I see in the notes I've used x minus one up to excite, so let me do that, that's maybe a bit tidier.

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So fi is constant on X.

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I'm minus one up to X I four. I equals one up and I let the value that it takes that b c i.

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So I suppose fi takes the value. See, I on that interval.

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Then we define. I off.I seems to be the sum from I was once and.

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Of CII times x minus X on minus one.

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So let me compute what is, for this example, function here.

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So here there are is to I guess let's just carefully write it down x nought equals not x one equals one x to equals two.

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And so I should b c one x one minus x nought plus c to x to minus x one.

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So what see, one, that's three. So that's three times one what?

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See, two. That's one one times one, which is equal to four.

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So if this function is for the what's the point of this definition?

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Well, hopefully we would agree that four is the only reasonable value of the integral of this function that I've drawn here.

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If somebody said to you, what is the integral of this function, the area under this function?

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I haven't defined it yet, but I want you to tell me as a sensible guess as to what it should be, I think the only possible thing you could say is for.

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So let me make some remarks. So the first remark is that I have fires what the integral of that function should be.

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Where should means, according to any reasonable intuition that you might have?

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But we haven't defined the interior yet. And the other remark I want to make, which is, I think again, something that all of you regard is intuitive,

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but it's perhaps a little bit less obvious is that the values of fight at the point I make absolutely no contribution to what this includes.

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So note that EFI is totally insensitive to the value of PHI at the point at the politician points.

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Exile, and I think that's something that is reasonably intuitive, I mean,

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they occupy areas zero those points and so they shouldn't make any contribution to the integral if the definition of integral is a sensible one.

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And there is a small subtlety to this definition.

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And the subtlety is that I've written ify as if this is a quantity that can be defined for any step function.

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But the definition uses the underlying politician pay. So the definition of EFI seems to.

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Depend on the underlying politician, P, however, that is an illusion.

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So why is that an illusion? Well, if I got really sort of overly precise and wrote, I have five comma p instead.

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So if we right eye of fi semicolon p for the quantity written above.

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Well, then the first thing one can see is that if I pass to a sub to a refinement of pay, it won't change the value.

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So I of five primed is equal to I of five P for any refinement.

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Of P. So why is that? Well, again, an example, I think it's pretty obvious if I passed a refinement, so I put the point three halves in here.

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I think it's reasonably obvious that that's not going to change the value that will still be for so now be something.

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Three. Plus, a half plus a half, which is four, so you could, if you want, write down a really careful proof of that, but I think that's pretty clear.

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And then if you've got two politicians, P1 and P2, you can pass to a common refinement if both of them.

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And so the values of those have got to be equal as well. So now if P1, P2 or any two politicians relative to which fires adapted.

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Take a common refinements of both of them. So let P be a common refinement of both of them.

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Common refinement of both, and then well, I have fi P. One is I have five, he is I have five two.

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And so it doesn't actually matter which politician I take and therefore the definition of I is well defined.

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So that's the choice of politician. Choice of politician is immaterial.

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And therefore, I have it is indeed well-defined. OK, one more little comment on this section, which is a in fact.

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And this is the fact that I is a linear functional on the space of step function, so I,

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from all step of AB to the rails is a linear functional, which is just a fancy way of saying.

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What basically that is. I mean, it's a linear map between vector spaces, so I of Lambda Phi one plus Ne Phi two is equal to lambda.

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I have five one plus mu higher Phi two.

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And the proof. Well, it's basically the same as the previous one point one if you pass to a common refinement.

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So if five everyone's adapted to partition one and five to two to pursue a common refinement of both.

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So they're adapted to the same partition and then it's just obvious so by passing to a common refinement.

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We may assume. That's five one and five two are adapted to the same politician pay.

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And then it's, I think, reasonably obvious. OK, so that is the definition of I, and now we can finally define what the integral is.

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So this will be another new section. So definition of the integral.

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So the way not now I'm going to look at arbitrary functions.

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And I'm going to try and assign some meaning to what they're integral over the interval AB should be.

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So I think this is definitely a point at which we're doing something different to what you've seen, probably at school,

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when is used to being given a function f as a kind of formula, maybe a polynomial or as a trigonometric function or some rational function.

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And you integrate it using some rules. But now we're just taking an abstractly defined function that may very well not be given by any useful formula.

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And we're going to try and say what's meant by its integral.

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And the way we would do that is to sandwich f between step function, so here's a picture of what we're going to try and do.

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So this is maybe if it doesn't have to be continuous by any means, but I've drawn a continuous function.

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And what we're going to try and do is to put step functions above it.

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Like that? So a step function that sits above Earth is going to be called a major and by a major and.

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Five plus five plus.

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For all I mean, is a step function that sits above f point wise.

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So we mean a step function. With F of X less than a week, four to five plus of X for all X.

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And then, of course, a monument is the same concept, but below. So this and monument would be something like this to not be the same.

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So a moment five minus four f, we mean the same thing, a step function with No.

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Five minus of X is less than or equal to F of X for all X.

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And so I think the picture is maybe self-explanatory. So now we say, let's say the f is integral, if I can find monuments and measurements for it,

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whose eye values are arbitrarily close together and the mathematically formalised way of writing that is as follows.

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So we say. I say the EFF is integral for.

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So I guess I'll say women insurable probably should be taboo win Super Bowl.

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If. The SOP over maintenance of I have five minus is equal to the over measurements of I have five plus.

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So where just to be clear, the SAP is over minus.

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Five minus four F and the inch is over measurements.

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I plus. And because this is just such an important definition or in the course.

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Let me just be completely clear about what Suffern doing here.

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What this means is that for any upsilon greater than zero,

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I can find a measurement five plus and a monument fund minus such that the difference of their values is that most asylum equivalently.

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So for any asylum crisis in Syria, there exists. So whenever I write this, I will mean monuments and major,

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and so there's a monument five minus and a measurement five plus with I have five plus minus, I find minus.

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Less than a week two asylum. OK, so that's what it means to be integral.

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What is the integral? So the integral is defined to be the common value of these two quantities.

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So the integral now we write that it's integral from A to B of F is defined to be.

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The common value. Star.

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So I think it's it's an intuitively appealing definition. We've defined somehow what the area is if these very basic stump functions called

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step functions and to try and define the integral of an arbitrary function.

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I'm going to try an approximated above and below by step functions in sandwich it between them, hopefully arbitrarily closely.

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OK, so a few little comments about this, keep that that.

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So the first comment is, in this case, we're only ever going to be working with bounded functions on A and B.

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Well,

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except right at the end where we'll talk about something called an improper integral but improper and schools are really just the kind of language,

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rather than something that's mathematically rigorous. So we'll always be dealing.

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We find it functions. F.

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So there's some universal uniform bound on a up to be.

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And. And what that means is that there is at least one major and one minor and four F.

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So in this case, if has a major hint.

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So, namely, the function that's just air everywhere. And it has a minor and.

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So that's five minus equals minus M everywhere. So that means at least up in the air, at least exist, so for banded functions.

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The infamous up exist.

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The other thing to say is that. If I have a minor inch in measurement, so if I've got five minus and five plus.

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Then the value of the monument is bounded above by the value of the measurement.

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So if five minus and five plus are minor into measurement. Then I of five minus is less than or equal to I have five plus.

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And the reason for this is that because of the definitions of minorities in measurement. Five minus is bounded above by five plus point was.

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And so if you pass to a common politician relative to which both of these are adopted,

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it's then just obvious that their values satisfy the same inequality.

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So passing to a politician, a p relative to which fi minus and five plus are both adopted.

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It's clear this this inequality. So the fact that there are values I have five minutes is less than a week to, I have five plus is clear.

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And so what this means. Because that's true for all minorities, five miners and all major,

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it's five plus then the left hand side here is always at most the right hand side here.

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So it's up over five minus five minus is at most over five plus I have five plus.

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And I guess that also means that if the function is insurable. Which is not always the case, but if essential, could.

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Then I of five minus is less than or equal to the integral is less than eight because I have five plus for any five minus and five plus.

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OK, so I've just written down a load of more or less obvious facts. It is no rental arithmetic.

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Is it in fact? Ten fifty seven. Oh,

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that's a shame because I was going to I was going to give an example of a

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function that's integral that isn't just a step function that I don't have time.

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So I think we'll leave it there for today, and next time, I will at least give an example of a function that's insurable.

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Also, the function that is not insurable. And then we'll start proving a few more substantial theorems.