Pre-image $\sigma$-algebra

Pre-image $\sigma$-algebra

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

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Definition

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)

Claim: $(X,\mathcal{A})$ is indeed a $\sigma$-algebra

Proof of claims

[Expand]

Claim 1: $(X,\mathcal{A})$ is indeed a $\sigma$-algebra

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See also

• Trace $\sigma$-algebra

OLD PAGE

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

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

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TODO: Measures Integrals and Martingales - page 16

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