Measurable Mappings to Product Measurable Space

Theorem

Let $\struct {X, \Sigma}$, $\struct {X_1, \Sigma_1}$ and $\struct {X_2, \Sigma_2}$ be measurable spaces.

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

Let $\pr_1: X_1 \times X_2 \to X_1$ and $\pr_2: X_1 \times X_2 \to X_2$ 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:

$\pr_i \circ f: X \to X_i$ is $\Sigma \, / \, \Sigma_i$-measurable, for $i = 1, 2$

Proof

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Sources

• 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $13.10 \ \text{(ii)}$