Preimage [ilmath]\sigma[/ilmath]algebra
Preimage [ilmath]\sigma[/ilmath]algebra  
[math]\{f^{1}(A')\ \vert\ A'\in\mathcal{A}'\}[/math] 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]. 
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 preimage [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 impliessubset relation)
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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See also
References

OLD PAGE
Let [ilmath]f:X\rightarrow X'[/ilmath] and let [ilmath]\mathcal{A}'[/ilmath] be a [ilmath]\sigma[/ilmath]algebra on [ilmath]X'[/ilmath], we can define a sigma algebra on [ilmath]X[/ilmath], called [ilmath]\mathcal{A} [/ilmath], by:
 [ilmath]\mathcal{A}:=f^{1}(\mathcal{A}'):=\left\{f^{1}(A')\vert\ A'\in\mathcal{A}'\right\}[/ilmath]
TODO: Measures Integrals and Martingales  page 16