I had some difficulty understanding Theorem 11.33 of Rudin's Principles of Mathematical Analysis. It has to do with the following question:
Suppose $L$ and $U$ are measurable functions on $X$. If $L(x)\le f(x) \le U(x)$, and $L(x)=U(x)$ almost everywhere, does this imply that $f$ is measurable?
Does the answer change if $X$ is a real interval $[a,b]$ and the measure is Lebesgue measure?
How do we prove these? (The answer to the second question is yes according to Rudin, and is used in the proof of Theorem 11.33, but it's not evident to me why this is true.) Thanks a lot!