Prove that the graph $f$ is measurable if $f$ is measurable.

Question

I know it's a classic question. It's that I want to prove using a specific approach.

I found in one or two hints one of the statements I want to use. The problem is, in these hints, this claim is not proved.

My approach is:

• Show that the function $(x,y) \mapsto f(x) - y$ is measurable.

• Conclude that graph(f) is inverse image of (a measurable) set $\{0\}$.

• Use Fubbini Theorem for prove that $m(graph(f)) = 0$.

My problem is to prove the first step. Can someone help me?

---

$m$ is the Lebesgue measure.

Answer 1

Try showing that the sum of measurable functions is measurable, and that $(x,y)\mapsto f(x)$ and $(x,y)\mapsto -y$ are measurable..

Answer 2

Your map is the sum of $f\circ \pi_X$ and $-\pi_Y$, each of which is measurable.