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## 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, [tex] \{ X_{\alpha} \}_{\alpha \in A}[/tex], and we have

[tex] X = \prod_{x \in \alpha}^{} X_{\alpha} [/tex]

and

[tex] \pi_{\alpha} : X \to X_{\alpha} [/tex]

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

{[tex] \pi_{\alpha}^{-1}(E_{\alpha}) : E_{\alpha} \in M_{\alpha}, \alpha \in A [/tex]}.

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 [tex] E_{\alpha} [/tex] from some [tex] M_{\alpha}[/tex] 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 [tex] E_{\alpha} [/tex] inside every [tex] M_{\alpha} [/tex]. 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 [tex] \pi_{\alpha}^{-1}(E_{\alpha}) [/tex] would look like just for a single [tex] E_{\alpha} [/tex].

## The Attempt at a Solution

I tried to take an example like this, Consider:

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

Is it [tex] E_{\alpha} \times R_{2} [/tex]?

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.