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

From Maths
Revision as of 23:53, 2 August 2015 by Alec (Talk | contribs) (Created page with "'''Title:''' A map, {{M|f:(A,\mathcal{A})\rightarrow(F,\mathcal{F})}}, is {{M|\mathcal{A}/\mathcal{F} }} measurable ''iff'' for some generator {{M|\mathcal{F}_0}} of {{M|\math...")

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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

  1. ↑ 1.0 1.1 Probability and Stochastics - Erhan Cinlar



Retrieved from "http://wiki.unifiedmathematics.com/index.php?title=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&oldid=1139"