real analysis - Does $lim_{nrightarrow infty} int_X f_n - int_X fgt 0$ implies that convergence of $f_n$ to $f$ a.e. fails?

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 $f\ a.e.$?

Answer

I assume you want to conclude that $\mu(\{x: f_n \not\to f\}) > 0$. The answer is NO.

Consider $$f_n = \begin{cases} n & x \in [0,1/n)\\ 0 & \text{otherwise} \end{cases}$$ and $$f = 0$$