Pre-image [ilmath]\sigma[/ilmath]-algebra

 Pre-image [ilmath]\sigma[/ilmath]-algebra $\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\}$ is a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath] given a [ilmath]\sigma[/ilmath]-algebra [ilmath](X',\mathcal{A}')[/ilmath] and a map [ilmath]f:X\rightarrow X'[/ilmath].
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Add to sigma-algebra index, link to other pages, general expansion. Needs to be exemplary as a lot of search traffic enters here.
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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:

• $\mathcal{A}:=\left\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\right\}$

We can write this (for brevity) alternatively as:

Claim: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra

Proof of claims

Claim 1: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra

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Should be pretty easy, it's just showing the definitions