A map from two sigma-algebras, A and B, is measurable if and only if for some generator of B (call it G) we have the inverse image of S is in A for every S in G

Title: A map, $f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})$, is $\mathcal{A}/\mathcal{F}$ measurable iff for some generator $\mathcal{F}_0$ of $\mathcal{F}$ we have $\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}]$

Statement

A map, $f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})$, is $\mathcal{A}/\mathcal{F}$ measurable iff for some generator $\mathcal{F}_0$ of $\mathcal{F}$ we have $\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}]$

Proof

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TODO: See ref page 6

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