Difference between revisions of "Pre-image sigma-algebra/Definition"
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(Created page with "<noinclude> ==Definition== </noinclude>Let {{M|(X,\mathcal{A}')}} be a algebra}} and let {{M|f:X\rightarrow X'}} be a map. The ''pre-image {{sigm...") |
m (Rephrasing slightly) |
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==Definition== | ==Definition== | ||
| − | </noinclude>Let {{M| | + | </noinclude>Let {{M|\mathcal{A}'}} be a [[sigma-algebra|{{sigma|algebra}}]] on {{M|X'}} and let {{M|f:X\rightarrow X'}} be a [[map]]. The ''pre-image {{sigma|algebra}} on {{M|X}}''{{rMIAMRLS}} is the {{sigma|algebra}}, {{M|\mathcal{A} }} (on {{M|X}}) given by: |
* {{MM|1=\mathcal{A}:=\left\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\right\} }} | * {{MM|1=\mathcal{A}:=\left\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\right\} }} | ||
We can write this (for brevity) alternatively as: | We can write this (for brevity) alternatively as: | ||
Latest revision as of 18:38, 1 April 2016
Definition
Let [ilmath]\mathcal{A}'[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X'[/ilmath] and let [ilmath]f:X\rightarrow X'[/ilmath] be a map. The pre-image [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath][1] is the [ilmath]\sigma[/ilmath]-algebra, [ilmath]\mathcal{A} [/ilmath] (on [ilmath]X[/ilmath]) given by:
- [math]\mathcal{A}:=\left\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\right\}[/math]
We can write this (for brevity) alternatively as:
- [math]\mathcal{A}:=f^{-1}(\mathcal{A}')[/math] (using abuses of the implies-subset relation)
