Proposition 15.5.1. Let $(X, \cm)$, $(Z, \cn)$ be measurable spaces, $\seqf{Y_j}$ separable metrisable topological spaces, $F: \prod_{j = 1}^{n} Y_{j} \to Z$ be a $(\cb_{\prod_{j = 1}^n Y_j}, \cn)$-measurable function. For any $\seqf{f_j}$ where for each $1 \le j \le n$, $f_{j}: X \to Y_{j}$ is $(\cm, \cb_{Y_j})$-measurable, the composition
\[X \to Z \quad x \mapsto F(f_{1}(x), \cdots, f_{n}(x))\]
is $(\cm, \cn)$-measurable.
Proof. By Proposition 15.2.3, $\cb_{\prod_{j = 1}^n Y_j}= \bigotimes_{j = 1}^{n} \cb_{Y_j}$, so
\[X \to \prod_{j = 1}^{n} Y_{j} \quad x \mapsto (f_{1}(x), \cdots, f_{n}(x))\]
is $(\cm, \cb_{\prod_{j = 1}^n Y_j})$-measurable. Therefore the composition is $(\cm, \cn)$-measurable.$\square$