So good morning and welcome to lecture five. Since you told me that you didn't

do measure theory in your mathematics courses so far, we'll have a short

interlude on that today and tomorrow we'll also look at Lebesgue integrals.

We do this for two reasons so we'll have a short recap of basic notions of

measure theory without proofs. That has two reasons. First of all we spend too

much time on it if I provide all the proofs and the proofs are not that

difficult and hence they will go to a problem sheet. The central proofs will go

to the problem sheet but that will be good practice. Short recap of the

basic notions without proofs for two reasons. The chief reason from the

abstract point of view is that the spectral theorem for self adjoint

operators we're going to prove in the next few lectures requires, as we

already saw from the axioms of quantum mechanics, the notion of projection

valued measures. Projection valued and the prediction valued is just a small

modification. Projection valued measures and unless we know what a measure is and

what a Borel measure is we're lost there and I rather explain it than to refer

you to the literature because that can be quite intimidating if you start at a

wrong point. And the second thing is that the most commonly emerging infinite

dimensional separable Hilbert space, now you say hang on this isn't there up to

unitary equivalence only one such? Yes there is but then separable but then it

depends on how you represent in order to make good use of it in physics. The most

commonly emerging infinite dimensional separable Hilbert space in quantum

mechanics is the Hilbert space L2 on some Rd, often d equals 3. And

informally speaking, that's not entirely correct, this is called these are all

square integrable functions on Rd. Well not quite because you have to take

equivalence classes of all square integrable functions because if two

square integrable functions are the same almost everywhere, what does that mean?

Well we'll see. Then they belong to the same equivalence class and then they're

counted as the same element of this. That's the first thing. And the second

thing if I say square integrable it is not square integrable with respect to the

Riemann integral but with respect to the Lebesgue integral and that makes all the

difference. You probably did in mathematics for physicists the number

of theorems telling you under what circumstances you can exchange integrals

and limits or integrals and sums and so on and you know that if you use the

Riemann integral you have rather strong conditions like uniform convergence and

stuff like this. Well the Lebesgue integral heals this. The Lebesgue integral is a

very proper notion of an integral which for Riemann integrable functions agrees

with them but which no longer poses such strong conditions from point of view of

Lebesgue theory virtually none on the functions in order to be able to

exchange limits and integrals. So that is what the physicists say for reasonably

well-behaved functions. Well you no longer even need to say this you say for

these and those functions without this rather uninsightful qualification of

being reasonably behaved. So the Lebesgue integral is the physicists integral but

it's also the mathematicians integral because this space equipped

with its inner product blah blah blah is only complete if you use the Lebesgue

integral. It wouldn't work with the Riemann integral okay. It simply won't.

It's not the Hilbert space then. Okay so there is no way in quantum mechanics

to avoid the Lebesgue integral. Everything else is child's play okay. So we're going

to do that and what we do today is the preparation for that. So we start

with section one general measure spaces. Now the first definition we provide is

that of a measurable space also called a sigma algebra and it looks a little like