Let $f:\mathbb R\to\mathbb R$ a measurable function. prove or disprove that $E=\{(x,\alpha)\mid 0\leq \alpha< |f(x)|\}$ is measurable.
I know it's measurable, but I really have no idea how to prove it.
P.S: Does $E=\{(x,\alpha)\mid 0\leq \alpha\leq |f(x)|\}$ still measurable or not ?
1) Let $Q$ measurable. If $f=\boldsymbol 1_Q$, then $E=(Q^c\times \{0\})\cup(Q\times [0,1])$ which is measurable.
2) If $f$ measurable, there is a sequence $\{\varphi_n\}$ s.t. $\varphi_n\nearrow |f|$. Let $$E_n=\{(x,\alpha)\mid 0\leq \alpha < \varphi_n\}.$$ We have that $E_{n}\subset E_{n+1}$ and that $$E=\bigcup_{i=1}^\infty E_n.$$ By 1) all the $E_n$ are measurable, and thus $E$ is measurable.
For your PS, the answer is yes.