Let $(X,\mathcal{M})$, $(E,\mathcal{E})$, and $(G,\mathcal{G})$ be measurable spaces. Let $f: X \to E$ be $\mathcal{M}/\mathcal{E}$-measurable, and let $g: X \to G$ be $\mathcal{M}/\mathcal{G}$-measurable. Equip $E \times G$ with the product $\sigma$-algebra
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\begin{align*}
\mathcal{E} \otimes \mathcal{G} := \sigma(\{A \times B : A \in \mathcal{E}, B \in \mathcal{G}\}).
\end{align*}
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Define $h: X \to E \times G$ by $h(x)=(f(x),g(x))$ for every $x \in X$. Then $h$ is $\mathcal{M}/(\mathcal{E}\otimes\mathcal{G})$-measurable.