Difference between revisions of "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"
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Revision as of 23:53, 2 August 2015
Title: A map, [ilmath]f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})[/ilmath], is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath] measurable iff 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]
Statement
A map, [ilmath]f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})[/ilmath], is [ilmath]\mathcal{A}/\mathcal{F} [/ilmath] measurable iff 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][1]
Proof
TODO: See ref[1] page 6
References
