Let
- $n\in\mathbb{N}$,
- $I_j\subseteq\mathbb{R}$, $1\le j\le n$, intervals,
- $I:= I_1\times\dots\times I_n\subseteq\mathbb{R}^n$,
- $f_j:I_j\to\mathbb{R}$, $1\le j\le n$, Lebesgue measurable,
- $g:I\to\mathbb{R}$ continuous,
- $h:I\ni(x_1,\dots,x_n)\mapsto g(f_1(x_1),\dots,f_n(x_n))\in\mathbb{R}$.
Then $h$ is Lebesgue measurable.
Do you know a textbook reference of this statement?