This question came up when I was studying for an analysis qualifying exam:
Suppose $f_n\geq 0$ for all $n\geq 1$, $f_n\rightarrow f$ a.e. on $[0,\infty)$ and there exists $M>0$ such that $$\sup_n\int_E f_n(x)\ \mathsf dx\leq M\mu(E)$$ for every measurable set $E\subset [0,\infty)$ with $\mu(E)>0$. Then $\mu\{x\in [0,\infty)\mid f(x)>M\}=0$. ($\mu$ denotes Lebesgue measure on $\mathbb{R}$.)
I have been trying to do something with Chebyshev's inequality, but I'm not sure I am on the right track. I would appreciate any pointers. Thanks in advance!