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Let $\{h_n\}_{n\in\mathbb{N}}$ be a sequence of positive functions with support in $\mathbb{R}$ such that $\int_{\mathbb{R}}h_n(x)dx \rightarrow 0$ as $n\rightarrow\infty$.

Question: Does this imply that $h_n \rightarrow 0$ as $n\rightarrow \infty$ almost everywhere?

EDIT: Suppose we further assume that $(h_n)$ is a sequence of continuous functions. Are there special properties around continuity that would ensure $h_n\rightarrow 0$?

user61038
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    See https://math.stackexchange.com/questions/138043/does-convergence-in-lp-implies-convergence-almost-everywhere – Robert Z Sep 29 '17 at 13:23
  • ... or have a look at the first counterexample in this answer: https://math.stackexchange.com/a/1170661/36150 – saz Sep 29 '17 at 13:50
  • Thankyou to the both of you, what happens if i were to restrict the problem to considering the case when the sequence $h_n$ is restricted to be a sequence of continuous functions? Would that remove particular degenerate cases? – user61038 Sep 30 '17 at 06:18

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