View source for 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

{{DISPLAYTITLE:A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})}}, is {{M|\mathcal{A}/\mathcal{F} }} measurable ''if and only if'' for some generator {{M|\mathcal{F}_0}} of {{M|\mathcal{F} }} we have {{M|\forall S\in\mathcal{F}_0[f^{-1}(S)\in\mathcal{A}]}}}}
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==Statement==
A [[map]] from a [[sigma-algebra|{{sigma|algebra}}]] {{M|(A,\mathcal{A})}} to another {{sigma|algebra}} {{M|(F,\mathcal{F})}}, {{M|f:A\rightarrow F}}, is [[measurable map|{{M|\mathcal{A}/\mathcal{F} }} measurable]] {{iff}} for some [[Generator (sigma-algebra)|generator]], {{M|\mathcal{G} }}, of {{M|\mathcal{F} }}<ref group="Note">Thus {{M|1=\mathcal{F}=\sigma(\mathcal{G})}}</ref> we have{{rMIAMRLS}}<ref name="PAS">Probability and Stochastics - Erhan Cinlar</ref>:
* {{M|\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}]}}
Which we may alternatively write (for brevity, see: [[abuses of the implies-subset relation]]) as:
* {{M|f^{-1}(\mathcal{G})\subseteq\mathcal{A} }}
==Proof==
==={{M|\implies}}: {{M|f:A\rightarrow B}} is {{M|\mathcal{A}/\mathcal{F} }}-measurable {{M|\implies}} for some generator {{M|\mathcal{G} }} of {{M|\mathcal{F} }} we have {{M|1=\forall S\in\mathcal{G}[f^{-1}(S)\in\mathcal{A}]}}===
* Let {{M|S\in\mathcal{G} }} be given
** Note that {{M|\mathcal{G}\subseteq\sigma(\mathcal{G})}}, so by the [[implies-subset relation]] we see {{M|S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})}}
** By [[measurable map|the definition of {{M|\mathcal{A}/\mathcal{F} }}-measurable]]:
*** {{M|\forall S\in F[f^{-1}(S)\in\mathcal{A}]}}
** Thus {{M|1=S\in\mathcal{G}\implies S\in\sigma(\mathcal{G})=\mathcal{F} }}
*** But as we've just seen, if {{M|S\in\mathcal{F} }} then {{M|f^{-1}(S)\in\mathcal{A} }}
* So {{M|f^{-1}(S)\in\mathcal{A} }}
This completes the proof
==={{M|\impliedby}}:===
{{Todo|See ref<ref name="PAS"/> page 6, also lemma 7.2 in<ref name="MIAMRLS"/>}}
==Notes==
<references group="Note"/>
==References==
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