Let $\{X_i\}_{i=1}^n$ be a collection of measurable spaces and $Y$ a measurable space. Is there a criterion that helps verify if a function $f:\Pi _{i=1}^nX_i\to Y$ is measurable?
I know that a function $f:Y\to \Pi _{i=1}^nX_i$ is measurable if, and only if, $\pi _j\circ f:Y\to X_j$ is measurable for all $j\in \{1,\cdots,n\}$ with $\pi_j:\Pi_{i=1}^nX_i\to X_j$ denoting the canonical projection. I want to know if there's a similar criterion for verifying the measurability of a function $f:\Pi _{i=1}^nX_i\to Y$.
Thank you for your attention!