Show that $x \mapsto f(x,g(x))$ is Borel measurable.

Question

Let $f:\mathbb{R}^2 \to \bar{\mathbb{R}}$ and $g: \mathbb{R} \to \mathbb{R}$ be Borel measurable functions. Show that $x \mapsto f(x,g(x))$ is Borel measurable.

Answer 1

We have: $$\mathcal B(\mathbb R^2)=\mathcal B(\mathbb R)\oplus\mathcal B(\mathbb R)$$

So that the function $(\mathsf{id},g):\mathbb R^2\to\mathbb R^2$ prescribed by $(u,v)\mapsto (u,g(v))$ is Borel measurable because $\mathsf{id}$ and $g$ are both Borel measurable.

Then composition $f\circ(\mathsf{id},g)$ will also be Borel measurable because it is a composition of Borel measurable functions.