Let $ f_{n}$ be a sequence of measurable non-negative functions such that $ \int f_{n} d\mu \rightarrow 0 , \mu$ is the measure of the considered measure space.
Does this imply that $f_{n} \rightarrow 0$ , $a.e.$ ?
I particularly want to know the answer to this question with respect to Lebesgue measure on the real line. But answer about general measure space is also fine.
