Measurable Mappings to Product Measurable Space

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Theorem

Let $\left({X, \Sigma}\right)$, $\left({X_1, \Sigma_1}\right)$ and $\left({X_2, \Sigma_2}\right)$ be measurable spaces.

Let $\Sigma_1 \otimes \Sigma_2$ be the product $\sigma$-algebra on $X_1 \times X_2$.

Let $\operatorname{pr}_1: X_1 \times X_2 \to X_1$ and $\operatorname{pr}_2: X_1 \times X_2 \to X_2$ be the first and second projections, respectively.


A mapping $f: X \to X_1 \times X_2$ is $\Sigma \, / \, \Sigma_1 \otimes \Sigma_2$-measurable if and only if:

$\operatorname{pr}_i \circ f: X \to X_i$ is $\Sigma \, / \, \Sigma_i$-measurable, for $i = 1, 2$


Proof


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