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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 $f\ a.e.$?

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    And you have to assume \delta>0 (and the measure >0) in order to make the assertion more interesting :)2012-10-28

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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 $f = 0$