Look at $(\mathbb{R^2},B(\mathbb{R^2}),m)$ with $E=\{(x,x)|x\in \mathbb{R}\}$
Prove that $E$ is measurable with $m(E)=0$.
My attempy: I used Tonelli's theorem: If i pick $f=1_E$, I get:
$m(E)=\int_{x\in \mathbb{R}} \int_{y\in \mathbb{R}}1_Edm_ydm_x=\int_{x\in \mathbb{R}} (\int_{y=x}1_Edm_y)dm_x=\int_{x\in \mathbb{R}}0dm_x=0$
Is this correct? Any help proving that E is measurable would be great as well.
Your argument is correct. The diagonal is a closed set, hence it is a Borel set.