This is the prelims course called Analysis three. And this course is about integration. So let me start by telling you a few things about what's going to be in the course. So essentially, this course is about making rigorous the notion of an integral. So how can we make the notion of an integral, rigorous cost control? And so basically, 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.
So we will make rigorous sense. 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, I'm sure you all aware of that the basic fact that integration and differentiation opposite operations, so integration and differentiation are opposites. So we'll formulate that really carefully and prove a couple of versions of it. And then we'll also look at some basic functions, which are best defined using integration.
So we'll look at the exponential and logarithm functions. And then we'll finish by looking at some things that you've not only known since secondary school, but since primary school, such as the fact that the circumference of a circle is two PI. So the circumference of the circle of Radius one is two PI and that the area of a circle of Radius one is PI.
So actually today to even define what the circumference or the area of a shape is, 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? So we'll also be carefully defining PI and showing that it's equal to these two things. When those two things are carefully defined. So these are some of the things that are going to be in the course of time. So another comment I want to make. It's a bit of an apology, really.
I'm apologising not for myself yet, but for the subjects of mathematics, really, so the integral that we're going to talk about in this course is not the right integral. So we'll be talking about. What's called the Raymond Integral? And it has the advantage that it's quite intuitive. It's relatively intuitive. And also relatively easy to define, so relatively, but it has some shortcomings, which we'll also be discussing.
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. And as a result of that, I think it's fair to say that professionals would always use what's called the LeBec integral. So professionals, whoever they are, tend to use the LeBec in school. 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?
Well, I think I've already mentioned it's relatively intuitive and easy to define, but thankfully whenever this remittance bill exists, it's equal to the baggage. So we are actually talking about a subset of the the kind of full theory that professional mathematicians would use. So this is a little bit like having to learn to crawl before you can learn to walk. Now, an apology that's more of a personal apology, but it's not really an apology, just a comment.
I'm actually not going to talk about the remaining screw. I'm actually going to talk about something called the taboo in school. And the reason I'm going to do that is that I find this the the best way to develop the theory. I find it just the most intuitive way to talk about the theory. And it's then a theorem that will prove later in the course that the top goal is the same as the remaining school.
So this turns out to be the same. As the minutes grow, but that's a theorem that has to be proven. 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, I would urge you to use the utmost caution if you look at any books about this subject early in the course, so you don't really need to because there are four lecture notes for the course on the internet.
But different books take subtly different approaches, and it will get very confusing, at least to begin with if you try and compare what I'm talking about with some books. 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, OK, so I think that's what I would say by way of introduction. So let's make a start on some actual mathematics. So the first chapter is about step functions and the integral.
So as I'm sure you all aware, the integral, whatever it is of a function, is supposed to be kind of measuring the area under the graph of that function, whatever that means. 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 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.
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, but one could talk about complex value functions without too much more difficulty. So this is called a step function. 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. I'm such that fight is constant on the open interval between X Y and Z plus one.
For between zero and minus one. So that's all it is. 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. So we don't care. What value? Fire takes at the south points outside. So just to make absolutely clear that we know what we're talking about. Let me give an example if I take, let's say, eight equals zero and vehicles two. So the function fly from zero up to two to the rear is defined as follows.
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. Sorry, X equals one. I'm one. If Pax lives between one and two and then minus six, if X equals two. So that's a step function and we could try and draw the graph of that just to really belabour the point. So here is supposed to be some axes that zero this one, that's two. And then this function is going to look a bit like this, so it will be 10 zero.
Three, on this open interval, there are minus seven down there. One here. And then minus six down here. So that's the graph of that step function, so that's what they all look like. We call a sequence of points that splits the interval AB into finally many parts that's called a partition. So a sequence of points. A equals x nought, that's three x one B is called a partition. Kelly P of the INS for Abby. And so we say that the step function is adapted to this particular politician.
OK, so hopefully everybody is pretty comfortable with this definition, I've gone over it quite slowly because it's going to be absolutely fundamental in the course. We'll just be talking about step functions over and over again because they're what's used in defining the integral. And just before stating a simple lemma, I want to wait one more definition, which is a very natural one. And this is the notion of a politician being a refinement of another politician.
And that's simply that you if you start with a politician such as zero one two here, then a refinement of it comes by just putting a few more points exi and keeping the original ones. So definition a politician primed. So that's going to consist of points Exide primed up to X and prime primed, and that's a refinement of P. If every X prise is an X, so if every X I primed is an X J. So that's the notion of refinement. It says clock on console is wrong.
Is there a clock that's actually right anywhere? Does anyone have an opinion on what the time is right now? The 10 19. OK, so I'm going to make the assumption that this clock is wrong, but sort of consistently wrong. That's 17 minutes fast. So I'm going to stop the lecture at about 11 12, according to this clock. OK, let's record. A simple lemma about politicians.
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. So suppose that fi is a step function adapted to some politician p then if I take a refinement of p p primes. Then fire is also a step function adapted to be primed. Second, if I've got two different politicians, then there's a common refinement of both of them.
So if I've got two politicians of the IDs for Abby, then there's another politician that refines them both. Common refinement. Pay. And then finally, just some. Closure properties of the space of step functions. If five, one and five two step functions. Then so are. Five, one plus five to scale and multiples, and then also things like the maximum nine five one five two. So this is this basically nothing to prove here? I'll make a remark about point three.
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. Point two is also basically obvious. You just take the points defining the position p one and the points defining the partition p two and throw them all in together to get your new set of partitioning points. And then three does deserve a little bit of a remark because these step functions may well be adapted to different politicians.
So I'm going to just say that one and two are pretty obvious. And then three? Well, I should at least say which politicians are going on here, so I suppose that PHI is adapted to PI. I equals one two. 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. So by passing to a common refinement. P. I can assume.
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, then there some is constantly scale multiple is constant in the max and the constant. So then this is obvious. Just looking at each of the sub intervals in the partition. 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,
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. 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. It's indicator function. One Sub X is the function of taking values. Value one for X and X and zero. If X is not an X. That's quite a standard definition.
And then the Lima number one point two is just the statement that the step functions are the same thing as the vector space spanned by indicator functions of intervals. So the space of step functions on a B equals the space of linear combinations, finite linear combinations, of course. Of indicator functions of intervals.
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, I think that is obvious from the definition of step function. So the indicator function of an interval. This is a step function, and I think that's obvious. And therefore, by the last lemma, a linear combination of indicator functions of intervals is also a step function. Hence, by Lemma 1.1, the third part, so is any linear combination.
So that's one direction, and then for the other direction, I need to know that a step function is a linear combination of indicator functions of intervals. So conversely, a step function is adapted to a particular politician. So P. Well, it's by definition it is constant on the open intervals, excite up to excite plus one. And so it's a linear combination. Of while the indicator functions of the open intervals. And. The trivial intervals, which are just the points, exi.
So notice that I was quite careful not to say whether my intervals were open or closed or half open or half closed, and so it's important here that I've allowed both open intervals and these closed intervals that consists of just one point. So let's just make that completely clear. Open interval. And this is a rather trivial closed interval. So that's a proof of dilemma.
So occasionally, this space of step functions, which I've not described in two different ways, just straight the basic definition. And as the linear span of indicator functions of intervals sometimes but not that often given the name. So sometimes we write our sub step of AB for the facts space of step functions. On Abe. So let me test my clock hypothesis is the time now 10 30. Good. So general comments about this course is one of these bits of maths where.
Once one has to go through a lot of very slightly tedious, really and quite simple lemon. 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. 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. So that section was all about just what a step function is.
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. And so this section is called I of a step function. 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, I'm going to assign a quantity called I, which I secretly know is going to be the integral, but I'm not going to allow myself to call it that just yet. So let me define what I'm talking about.
So suppose I have a step function, so let five be a step function. Adapt it to. A politician, P. Between the on on a up to B and let us suppose so, by definition, it's constant between X, Y and Z. 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. So fi is constant on X. 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. So I suppose fi takes the value. See, I on that interval.
Then we define. I off.I seems to be the sum from I was once and. Of CII times x minus X on minus one. So let me compute what is, for this example, function here. 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. And so I should b c one x one minus x nought plus c to x to minus x one. So what see, one, that's three. So that's three times one what? See, two. That's one one times one, which is equal to four.
So if this function is for the what's the point of this definition? Well, hopefully we would agree that four is the only reasonable value of the integral of this function that I've drawn here. If somebody said to you, what is the integral of this function, the area under this function? 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.
So let me make some remarks. So the first remark is that I have fires what the integral of that function should be. Where should means, according to any reasonable intuition that you might have? 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, 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.
So note that EFI is totally insensitive to the value of PHI at the point at the politician points. Exile, and I think that's something that is reasonably intuitive, I mean, 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. And there is a small subtlety to this definition. And the subtlety is that I've written ify as if this is a quantity that can be defined for any step function.
But the definition uses the underlying politician pay. So the definition of EFI seems to. Depend on the underlying politician, P, however, that is an illusion. So why is that an illusion? Well, if I got really sort of overly precise and wrote, I have five comma p instead. So if we right eye of fi semicolon p for the quantity written above. 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.
So I of five primed is equal to I of five P for any refinement. 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. I think it's reasonably obvious that that's not going to change the value that will still be for so now be something. 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.
And then if you've got two politicians, P1 and P2, you can pass to a common refinement if both of them. 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. Take a common refinements of both of them. So let P be a common refinement of both of them. Common refinement of both, and then well, I have fi P. One is I have five, he is I have five two.
And so it doesn't actually matter which politician I take and therefore the definition of I is well defined. So that's the choice of politician. Choice of politician is immaterial. And therefore, I have it is indeed well-defined. OK, one more little comment on this section, which is a in fact. And this is the fact that I is a linear functional on the space of step function, so I, from all step of AB to the rails is a linear functional, which is just a fancy way of saying.
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. I have five one plus mu higher Phi two. And the proof. Well, it's basically the same as the previous one point one if you pass to a common refinement. So if five everyone's adapted to partition one and five to two to pursue a common refinement of both. So they're adapted to the same partition and then it's just obvious so by passing to a common refinement.
We may assume. That's five one and five two are adapted to the same politician pay. 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. So this will be another new section. So definition of the integral. So the way not now I'm going to look at arbitrary functions. And I'm going to try and assign some meaning to what they're integral over the interval AB should be.
So I think this is definitely a point at which we're doing something different to what you've seen, probably at school, 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. 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. And we're going to try and say what's meant by its integral.
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. So this is maybe if it doesn't have to be continuous by any means, but I've drawn a continuous function. And what we're going to try and do is to put step functions above it. Like that? So a step function that sits above Earth is going to be called a major and by a major and. Five plus five plus. For all I mean, is a step function that sits above f point wise.
So we mean a step function. With F of X less than a week, four to five plus of X for all X. 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. So a moment five minus four f, we mean the same thing, a step function with No. Five minus of X is less than or equal to F of X for all X.
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, whose eye values are arbitrarily close together and the mathematically formalised way of writing that is as follows. So we say. I say the EFF is integral for. So I guess I'll say women insurable probably should be taboo win Super Bowl. If. The SOP over maintenance of I have five minus is equal to the over measurements of I have five plus.
So where just to be clear, the SAP is over minus. Five minus four F and the inch is over measurements. I plus. And because this is just such an important definition or in the course. Let me just be completely clear about what Suffern doing here. What this means is that for any upsilon greater than zero, I can find a measurement five plus and a monument fund minus such that the difference of their values is that most asylum equivalently.
So for any asylum crisis in Syria, there exists. So whenever I write this, I will mean monuments and major, and so there's a monument five minus and a measurement five plus with I have five plus minus, I find minus. Less than a week two asylum. OK, so that's what it means to be integral. What is the integral? So the integral is defined to be the common value of these two quantities. So the integral now we write that it's integral from A to B of F is defined to be.
The common value. Star. 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 step functions and to try and define the integral of an arbitrary function. I'm going to try an approximated above and below by step functions in sandwich it between them, hopefully arbitrarily closely. OK, so a few little comments about this, keep that that.
So the first comment is, in this case, we're only ever going to be working with bounded functions on A and B. Well, 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, rather than something that's mathematically rigorous. So we'll always be dealing. We find it functions. F. So there's some universal uniform bound on a up to be.
And. And what that means is that there is at least one major and one minor and four F. So in this case, if has a major hint. So, namely, the function that's just air everywhere. And it has a minor and. 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. The infamous up exist. 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.
Then the value of the monument is bounded above by the value of the measurement. 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. 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.
And so if you pass to a common politician relative to which both of these are adopted, it's then just obvious that their values satisfy the same inequality. So passing to a politician, a p relative to which fi minus and five plus are both adopted. 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.
And so what this means. Because that's true for all minorities, five miners and all major, it's five plus then the left hand side here is always at most the right hand side here. So it's up over five minus five minus is at most over five plus I have five plus. And I guess that also means that if the function is insurable. Which is not always the case, but if essential, could.
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. OK, so I've just written down a load of more or less obvious facts. It is no rental arithmetic. Is it in fact? Ten fifty seven. Oh, that's a shame because I was going to I was going to give an example of a function that's integral that isn't just a step function that I don't have time.
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. Also, the function that is not insurable. And then we'll start proving a few more substantial theorems.