Understanding Product Sigma Algebras: Definition, Notation, and Example

[Aerostd]

Homework Statement

I should mention beforehand that I do not come from a math background so I may ask some trivial questions.

I am reading the book "Real Analysis" by Folland for a course I am taking and am attempting to understand a definition of product sigma algebra. It is stated in the book that if we have an indexed collection of non-empty sets, \{ X_{\alpha} \}_{\alpha \in A}, and we have

X = \prod_{x \in \alpha}^{} X_{\alpha}

and

\pi_{\alpha} : X \to X_{\alpha}

the coordinate maps. If M_{\alpha} is a \sigma-algebra on X_{\alpha} for each \alpha, the product \sigma-algebra on X is the \sigma-algebra generated by

{\pi_{\alpha}^{-1}(E_{\alpha}) : E_{\alpha} \in M_{\alpha}, \alpha \in A}.Now the first time I read this definition(by first time I mean the thousandth time), I thought that one has to just pick one E_{\alpha} from some M_{\alpha} and take it's inverse map(pre image?) to get some collection of sets. Then generating a sigma algebra from that collection would yield the product sigma algebra. However, I was told that we have to take the inverse map of every E_{\alpha} inside every M_{\alpha}. I don't know how this is implied by the definition stated in the book but it makes more sense. Maybe I am not familiar with the notation because I would have expected some "for all" symbols in the definition somewhere.

Secondly, I wanted to think about what \pi_{\alpha}^{-1}(E_{\alpha}) would look like just for a single E_{\alpha}.

The Attempt at a Solution

I tried to take an example like this, Consider:

\{ R_{1}, R_{2} \}, the real numbers. Then X = R_{1} \times R_{2}. Suppose I have sigma algebras M_{1} and M_{2}. Now, What is \pi_{\alpha}^{-1} (E_{\alpha}) for some element in say M_{1}?

Is it E_{\alpha} \times R_{2}?

I don't know. Even this is just a guess. I'm hoping someone can give me some hints on understanding this since I am not able to easily find references on product sigma algebras.

 

[micromass]

Maybe I am not familiar with the notation because I would have expected some "for all" symbols in the definition somewhere.

Well, the "for all" symbol is implied by the notation. Let P be a certain property (I'll give examples later), then

\{x\in A~\vert~P(x)\}

means that you take all elements in A which satisfy P. An example:

\{n^2~\vert~n\in \mathbb{N}\}

By definition this means that you take all the squares in \mathbb{N}. Thus, you take all the elements in \mathbb{N}, and you square it. Another example:

\{\{x\}~\vert~x\in \mathbb{R}\}

means that you take all the x\in \mathbb{R}, and you take the singleton set of it.

So, your definition:

\{\pi^{-1}(E_\alpha)~\vert~E_\alpha\in M_\alpha, \alpha\in A\}

means that you take all the \alpha\in A and all the E_\alpha\in M_\alpha, and you calculate \pi^{-1}(E_\alpha).

Is it E_{\alpha} \times R_{2}?

This is correct! \pi^{-1}(E_\alpha)=E_\alpha\times R_2...

 

[Aerostd]

Thanks a lot. I guess now I will move on to the next page in the book. I don't know if I should open a new thread or not but here goes.

This is Proposition 1.3 in Folland. I am interested more in the notation than in the proof given in the book. The proposition is

If A is countable, then the product sigma-algebra is the sigma-algebra generated by \{ \prod_{\alpha \in A} E_{\alpha} : E_{\alpha} \in M_{\alpha} \}.

The proof given is this:

If E_{\alpha} \in M_{\alpha}, then \pi_{\alpha}^{-1} (E_{\alpha}) = \prod_{\beta \in A} E_{\beta} where E_{\beta} = X_{\beta} for \beta \notin \alpha.

On the other hand, \prod_{\alpha \in A} E_{\alpha} = \cap_{\alpha \in A} \pi_{\alpha}^{-1} (E_{\alpha})

Next he uses a lemma to prove this. I am more interested in the two equalities he uses.

Before anything I will ask the stupid question. Shouldn't one of the elements in M_{\alpha} be the empty set? That is there should be some E_{j} which will just be the empty set. Then the cartesian product of all elements in M_{\alpha} should be the empty set. Where am I goofing up?

Anyway again I consider my example where I have \{ R_{1}, R_{2} \}, the reals and then have M_{1} for R_{1} and M_{2} for R_{2}.

Now suppose I consider E_{1} in M_{1}. Then the first equality should be \pi_{1}^{-1} E_{1} = E_{1} \times R_{2}, which I can understand.

Next, in the second equality, the LHS I already asked about before. The RHS looks like it should be = E_{1} \times R_{1} \cap E_{2} \times R_{1} \cap E_{3} \times R_{1} \cap ..., but I don't see how this condenses to E_{1} \times E_{2} \times ... which I am assuming is the RHS.