True/false : If $f_n(x)\to 0$ almost everywhere , then $\int_{a}^{b}f_n(x)dx \to 0$

The usual counterexample is something like $$f_n(x) = \begin{cases}n & x \in (0, 1/n] \\ 0 & \text{otherwise}\end{cases}$$


Take for instance the sequence $f_n=n1_{[0,\frac{1}{n}]}$ on $[0,1]$

Tags:

Real Analysis

Measure Theory

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