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

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## Statement

A map from a [ilmath]\sigma[/ilmath]-algebra [ilmath](A,\mathcal{A})[/ilmath] to another [ilmath]\sigma[/ilmath]-algebra [ilmath](F,\mathcal{F})[/ilmath], [ilmath]f:A\rightarrow F[/ilmath], is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath] measurable if and only if for some generator, [ilmath]\mathcal{G} [/ilmath], of [ilmath]\mathcal{F} [/ilmath][Note 1] we have[1][2]:

• [ilmath]\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}][/ilmath]

Which we may alternatively write (for brevity, see: abuses of the implies-subset relation) as:

• [ilmath]f^{-1}(\mathcal{G})\subseteq\mathcal{A} [/ilmath]

## Proof

### [ilmath]\implies[/ilmath]: [ilmath]f:A\rightarrow B[/ilmath] is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath]-measurable [ilmath]\implies[/ilmath] for some generator [ilmath]\mathcal{G} [/ilmath] of [ilmath]\mathcal{F} [/ilmath] we have [ilmath]\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}][/ilmath]

• Let [ilmath]S\in\mathcal{G} [/ilmath] be given
• Note that [ilmath]\mathcal{G}\subseteq\sigma(\mathcal{G})[/ilmath], so by the implies-subset relation we see [ilmath]S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})[/ilmath]
• By the definition of [ilmath]\mathcal{A}/\mathcal{F} [/ilmath]-measurable:
• [ilmath]\forall S\in F[f^{-1}(S)\in\mathcal{A}][/ilmath]
• Thus [ilmath]S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})=\mathcal{F}[/ilmath]
• But as we've just seen, if [ilmath]S\in\mathcal{F} [/ilmath] then [ilmath]f^{-1}(S)\in\mathcal{A} [/ilmath]
• So [ilmath]f^{-1}(S)\in\mathcal{A} [/ilmath]

This completes the proof

### [ilmath]\impliedby[/ilmath]:

TODO: See ref[2] page 6, also lemma 7.2 in[1]

## Notes

1. Thus [ilmath]\mathcal{F}=\sigma(\mathcal{G})[/ilmath]

## References

1. Measures, Integrals and Martingales - René L. Schilling
2. Probability and Stochastics - Erhan Cinlar