5-21

Question 21: If \int_A f=0 for every measurable subset, prove that f=0 almost everyway.
Answer:
Note that 
\int_\mathcal{R}|f|=\int_{x>0}f-\int_{x\leq 0}f =0
in the case that if f\neq 0 in a nonzero measure set, this couldn't happen, this proves the theorem.