Difference between revisions of "Pre-image sigma-algebra/Definition"
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Revision as of 13:49, 18 March 2016
Definition
Let [ilmath](X,\mathcal{A}')[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra 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)
