Pre-image sigma-algebra/Definition

Definition

Let $(X,\mathcal{A}')$ be a $\sigma$-algebra 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)