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So good morning and welcome to lecture number six of the quantum theory course.

Today we'll consider the integration of measurable functions and we'll do this for a number of purposes.

The key definition we're going to provide today is that of a Lebesgue integral.

Now last time we had the Lebesgue measure and the Lebesgue measure was a very special measure on Rd.

Now the Lebesgue integral is not necessarily be performed with the Lebesgue measure.

So although it's also called the Lebesgue integral, it's just that of course Lebesgue worked on this kind of stuff.

So the Lebesgue integral is more general, of course you can have the Lebesgue measure of a measurable function,

an integrable function with respect to the Lebesgue measure.

So that can be the case but it's not necessarily the case.

So I would like to point that out from the beginning.

And the key application we're going to look at today for quantum mechanics is we'll give a proper definition

and some of the properties and why it's so important to have the Lebesgue integral.

We're going to define so-called LP spaces.

And roughly speaking those are the spaces of functions which are whose pth power,

or rather pth power of the absolute value is integrable.

But there are certain additional gymnastics one has to entertain in order to make this into a proper Banach space,

or in the case p equals 2, which is even more special than the other ones,

in the case p equals 2 the L2 space is even a Hilbert space,

whereas the others are only Banach spaces and the L2 spaces are the space of square integrable functions,

or which is only approximately described by the space of square integrable functions

that in quantum mechanics appear all over the place.

So that's our task for today, but before we can get there we have a number of definitions to make in order to get the concepts straight.

So the first section is on so-called simple functions, other people call them elementary functions.

They are an important construction tool in order to define the Lebesgue integral.

And so we need to look at them first, so as a technical tool we introduce the following definition.

A measurable function, let's call it s like simple, a measurable function on some set M into the real numbers,

but we will in fact only look at functions into the real numbers that are non-negative,

is called a simple function if s takes the entire domain into only a finite image.

So several remarks. Well first of all if I call this a measurable function what do I need to know here?

What do I need to know? What structure on M and what structure on R plus zero do I need?

We need measurable spaces.

We need measurable spaces, that's right. So I need a sigma algebra here and I need a sigma algebra here.

So implicit when I say measurable here, implicit in that is that I have chosen a sigma algebra sigma on here

and that I have chosen one here. Well this one here, this is free to be chosen, that depends on the application,

whatever set and whatever sigma algebra you have, but we have an agreement what we do on the real numbers.

What kind of sigma algebra do we choose there? Well we usually choose the Borel sigma algebra

and here so you could take that of the standard topology or if you wish you can first restrict to here,

so you have to induce topology on R plus zero, something of the type.

So a measurable function is called a simple function if, but now the actual definition of simplicity apart from

measurability is just that the domain, that the function takes only finitely many different values.

That's it. So it's truly a simple function, right. It must look somehow, well it's difficult to draw this M

because we don't know what type of set it is, so I make a double line, it's more like symbolic,

and here I have R plus zero starting here, so it must look somehow like, somehow like this, okay.

And then at some point this goes on forever, this goes on forever, well forever meaning as far as M extends.

So I have one, two, three, four different values, N equals four. Very, very simple.

Okay, so now an immediate implication of that definition, and that is actually why we need the measurability in here,

is that the pre-image of a simple, of a set of just one of the values S i with respect to the simple function,

well what is that? Well this here is one point in R plus zero which is equipped with this sigma algebra,

so clearly a set consisting of a single point is measurable in here. So this guy here is Borel measurable,