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Possible Duplicate:
Preimage of generated $\sigma$-algebra

I wish to prove the following:

"Let $X$ be a set and $\mathcal{A}$ a family of subsets of $X$, and $\Sigma_{\mathcal{A}}$ the $\sigma$-algebra of subsets of $X$ generated by $\mathcal{A}$. Suppose that $Y$ is another set and $f : Y \rightarrow X$ a function. Then $\left\{f^{-1} \left[E \right] : E \in \Sigma_{\mathcal{A}} \right\}$ is the $\sigma$-algebra of subsets of $Y$ generated by $\left\{ f^{-1} \left[A\right] : A \in \mathcal{A} \right\}$."

I understand that the $\sigma$-algebra of subsets of $Y$ generated by $\left\{ f^{-1} \left[A\right] : A \in \mathcal{A} \right\}$ is defined to be $\bigcap \left\{ \Sigma : \Sigma \text{ is a } \sigma \text{-algebra of subsets of }Y, \left\{ f^{-1} \left[A\right] : A \in \mathcal{A} \right\} \subseteq \Sigma\right\}.$

I'm not sure how this leads to the desired result, though. Any help much appreciated.

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    The answer to your question is here: http://math.stackexchange.com/questions/7881/preimage-of-generated-sigma-algebra2012-02-18

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I think the best approach will be to just show inclusion of the two sigma algebras in both directions.

I am studying this kind of material as well at the moment, so my answer needs considered with this caveat in mind. I m sure better answers will be provided in due course !

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    At least, this is the technical definition in my lecture notes; I can see there are some pre-images here, but they seem more specific than those in the problem here, so I'm not sure how it helps?2012-02-06