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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.

Community
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Harry Williams
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    The answer to your question is here: http://math.stackexchange.com/questions/7881/preimage-of-generated-sigma-algebra –  Feb 18 '12 at 23:54

1 Answers1

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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 !

Beltrame
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    Something to consider: What do you mean by "$f$ is a measurable function"? (Indeed, in a sense, that's precisely what you're trying to create here.) :) – cardinal Jan 24 '12 at 02:23
  • I think what you are saying is that from the way we define the domain of $\quad f \quad$ it will necessarily be a measurable function right ? Bit new to this material, so just wondering whether I understand you correctly. (post amended accordingly...) – Beltrame Jan 24 '12 at 02:32
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    Yes, essentially. – cardinal Jan 24 '12 at 02:44
  • I understand $f:D \rightarrow \mathbb{R}$ to be a measurable function if any, or equivalently all, of the following are true: (i) ${x : f(x) < a} \in \Sigma_D$ for every $a \in \mathbb{R}$; (ii) ${x : f(x) \leq a} \in \Sigma_D$ for every $a \in \mathbb{R}$; (iii) (i) ${x : f(x) > a} \in \Sigma_D$ for every $a \in \mathbb{R}$; (iv) (i) ${x : f(x) \geq a} \in \Sigma_D$ for every $a \in \mathbb{R}$, where $D \subseteq X$ and $\Sigma_D$ is the subspace $\sigma$-algebra of subsets of $D$. – Harry Williams Feb 06 '12 at 22:12
  • 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? – Harry Williams Feb 06 '12 at 22:17