Show that $f$ is measurable in $\Bbb R\ \iff$ the set $\{(x,y)\mid f(x)>y\}$ is measurable in $\Bbb R^2$
The problem doesn't seem very hard, at least on the surface for me. Forward direction is ok. $f(x)-y$ is measurable in $\Bbb R^2$ and so is $f(x)-y>0$. The backwards direction is, I think, harder. I'm not sure we can claim that each slice is measurable unless we know whether $\Bbb R^2$ has the completed measure or not. If it's incomplete, then, we know that slice of a measurable set is measurable and we're done. However, if it's the completion, I don't know what to do.