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

Statement

That the pre-image $\sigma$-algebra is indeed a $\sigma$-algebra.

Definition of the pre-image $\sigma$-algebra

Let $\mathcal{A}'$ be a $\sigma$-algebra on $X'$ and let $f:X\rightarrow X'$ be a map. The pre-image $\sigma$-algebra on $X$ is the $\sigma$-algebra, $\mathcal{A}$ (on $X$) given by:

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

We can write this (for brevity) alternatively as:

• $$\mathcal{A}:=f^{-1}(\mathcal{A}')$$ (using abuses of the implies-subset relation)

Proof

(Unknown grade)

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

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