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

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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][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:


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


  1. Measures, Integrals and Martingales - René L. Schilling