I've come across this problem as a part of another proof that I'm writing and I want to know if this is a right conclusion:
Let $X$ be a finite measure space and $\{f_n\}$ be a sequence of nonnegative integrable functions.
If I know: $\lim_{n\rightarrow \infty} \int_X f_n - \int_X f \geq \delta > 0,$ can I conclude that $\mu\{x: f_n \nrightarrow f\} > 0$ or in other words $f_n$ doesn't convege to $f\ a.e.$?