Pre-image [ilmath]\sigma[/ilmath]-algebra

Pre-image [ilmath]\sigma[/ilmath]-algebra

[math]\{f^{-1}(A')\ \vert\ A'\in\mathcal{A}'\}[/math] is a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X[/ilmath] given a [ilmath]\sigma[/ilmath]-algebra [ilmath](X',\mathcal{A}')[/ilmath] and a map [ilmath]f:X\rightarrow X'[/ilmath].

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Definition

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)

Claim: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra

Proof of claims

Claim 1: [ilmath](X,\mathcal{A})[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra

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See also

• Trace [ilmath]\sigma[/ilmath]-algebra

OLD PAGE

Let [ilmath]f:X\rightarrow X'[/ilmath] and let [ilmath]\mathcal{A}'[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra on [ilmath]X'[/ilmath], we can define a sigma algebra on [ilmath]X[/ilmath], called [ilmath]\mathcal{A} [/ilmath], by:

• [ilmath]\mathcal{A}:=f^{-1}(\mathcal{A}'):=\left\{f^{-1}(A')\vert\ A'\in\mathcal{A}'\right\}[/ilmath]

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TODO: Measures Integrals and Martingales - page 16

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