Proof that $f$ is integrable on $[0,1]^2$ if and only if $g$ is integrable on $[0,1]$

Question

Let $g: [0,1] \rightarrow \mathbb R$ be a measurable function and $f :[0,1]^2 \rightarrow \mathbb R$ such that: $f(x,y) = 2g(x) - 3g(y)$. Proof that $f$ is integrable on $[0,1]^2$ if and only if $g$ is integrable on $[0,1]$

Can anyone help me? I failed to reach anything meaningful that would lead me to the thesis ...

Answer 1

If $g$ is integrable the $f$ is a linear combination of two intergable functions.

If $f$ is integrable then Fubini's Theorem tells you that $\int f(x,y)dx$ is an integrable function of $y$. This shows that $g(y)$ is an integrable function of $y$.