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kalish1
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**Theorem:**
Let $(\Omega,\mathcal{F})$ be a measurable space and let $f:\Omega \rightarrow Y$ be a given function. Let $\mathcal{A}$ be a collection of subsets of $Y$.
If $f^{-1}(A) \in \mathcal{F}$ for every $A \in \mathcal{A}$, then $f^{-1}(A) \in \mathcal{F}$ for every $A \in \sigma(\mathcal{A})$.
**Questions**:
I understand that $\Omega$ is the underlying set and $\mathcal{F}$ is a class of subsets of $\Omega$ that is closed under finite unions and complementation (i.e., $\mathcal{F}$ is a field). But what is $Y$? And what is the purpose of defining $\mathcal{A}$ as a *collection of subsets* of $Y$, with $A$ an element in that collection? Moreover, what is the message of the theorem?
The notation is throwing me off.
This question has been crossposted on real analysis - What is this theorem about measurable functions saying? - Mathematics Stack Exchange
Let $(\Omega,\mathcal{F})$ be a measurable space and let $f:\Omega \rightarrow Y$ be a given function. Let $\mathcal{A}$ be a collection of subsets of $Y$.
If $f^{-1}(A) \in \mathcal{F}$ for every $A \in \mathcal{A}$, then $f^{-1}(A) \in \mathcal{F}$ for every $A \in \sigma(\mathcal{A})$.
**Questions**:
I understand that $\Omega$ is the underlying set and $\mathcal{F}$ is a class of subsets of $\Omega$ that is closed under finite unions and complementation (i.e., $\mathcal{F}$ is a field). But what is $Y$? And what is the purpose of defining $\mathcal{A}$ as a *collection of subsets* of $Y$, with $A$ an element in that collection? Moreover, what is the message of the theorem?
The notation is throwing me off.
This question has been crossposted on real analysis - What is this theorem about measurable functions saying? - Mathematics Stack Exchange