Pre-image sigma-algebra/Definition

Definition

Let [ilmath](X,\mathcal{A}')[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra 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)