Consider a locally bounded function $f: X \times W \rightarrow X$, where $X \subseteq \mathbb{R}^n$, $W \subseteq \mathbb{R}^m$, such that
for all $x \in X$ the function $w \mapsto f(x,w)$ is (Borel) measurable;
Consider a locally bounded, (Borel) measurable, function $g: W \rightarrow X$.
Say if the function
$ (w,v) \mapsto f( g(w), v ) $
is (Borel) measurable as well.