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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| − | + | {{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}]}}}} | |
| + | {{Stub page}} | ||
| + | __TOC__ | ||
==Statement== | ==Statement== | ||
| − | A map | + | 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== | ==Proof== | ||
| − | {{Todo|See ref<ref name="PAS"/> page 6}} | + | {{Todo|See ref<ref name="PAS"/> page 6, also lemma 7.2 in<ref name="MIAMRLS"/>}} |
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
==References== | ==References== | ||
<references/> | <references/> | ||
| + | {{Measure theory navbox|plain}} | ||
{{Theorem Of|Measure Theory}} | {{Theorem Of|Measure Theory}} | ||
Revision as of 21:13, 17 March 2016
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Contents
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
TODO: See ref[2] page 6, also lemma 7.2 in[1]
Notes
- ↑ Thus [ilmath]\mathcal{F}=\sigma(\mathcal{G})[/ilmath]
References
- ↑ 1.0 1.1 Measures, Integrals and Martingales - René L. Schilling
- ↑ 2.0 2.1 Probability and Stochastics - Erhan Cinlar
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