Pre-image sigma-algebra/Proof of claim: it is a sigma-algebra

Statement

That the pre-image [ilmath]\sigma[/ilmath]-algebra is indeed a [ilmath]\sigma[/ilmath]-algebra.

Definition of the pre-image [ilmath]\sigma[/ilmath]-algebra

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] 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)

Proof

(Unknown grade)

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