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Let $\Sigma$ be a $\sigma$-algebra over $\mathbb R$ and $\mathcal A \subset \mathcal P(\mathbb R)$. Let also $f: \mathbb R \to \mathbb R$ be any function.

If $\mathcal A$ generates $\Sigma$, is it true that $\widetilde{f^{-1}}(\mathcal A)$ generates $\widetilde{f^{-1}}(\Sigma)$? That is, do these symbols commute:

$\sigma(\widetilde{f^{-1}}(\mathcal A)) = \widetilde{f^{-1}}(\sigma(\mathcal A))\quad?$

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    Possible duplicate: http://math.stackexchange.com/questions/7881/preimage-of-generated-sigma-algebra (although I am late :) )2011-11-23

1 Answers 1

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For any set-theoretic function $f\colon A\to B$, the inverse image function $f^{-1}\colon\mathcal{P}(B)\to\mathcal{P}(A)$ given by $f^{-1}(Y) = \{a\in A\mid f(a)\in Y\}$ is extremely well-behaved relative to set operations. In particular, for all $X,Y\subseteq B$ and all families $\{X_i\}\subseteq \mathcal{P}(B)$, $\begin{align*} f^{-1}(X\cup Y) &= f^{-1}(X)\cup f^{-1}(Y),\\ f^{-1}(X\cap Y) &= f^{-1}(X)\cap f^{-1}(Y),\\ f^{-1}(\cup X_i) &= \cup f^{-1}(X_i),\\ f^{-1}(\cap X_i) &= \cap f^{-1}(X_i),\\ f^{-1}(X-Y) &= f^{-1}(X) - f^{-1}(Y),\\ f^{-1}(X^c) &= (f^{-1}(X))^c,\\ f^{-1}(X\triangle Y) &= f^{-1}(X)\triangle f^{-1}(Y) \end{align*}$ where $\triangle$ is the symmetric difference. As such, the inverse image of a family that generates a $\sigma$-algebra will generate the inverse image of the $\sigma$-algebra generated: you can justify the details by looking at the "bottoms-up" description of the $\sigma$-algebra generated by a family that appears in Asaf's answer to this question.

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    @xFioraMstr18 4: See [Example of the good set principle](https://math.stackexchange.com/questions/2064953/example-of-the-good-set-principle) AND [Proving that a class of good sets forms a $\sigma$-field.](https://math.stackexchange.com/questions/2304152/proving-that-a-class-of-good-sets-forms-a-sigma-field) AND [$\sigma$-algebra produced by a subclass of a class.](https://math.stackexchange.com/questions/1651937/sigma-algebra-produced-by-a-subclass-of-a-class) AND [p. 11 here](http://math.utoledo.edu/~dwhite1/d_6800/measurable.pdf) AND [p. 10 here](https://ee.stanford.edu/~gray/arp.pdf).2017-11-13