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Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function such that

$$\left|\int_I h\right|\leq c \sqrt{|I|}$$

for each interval $I$. Then $h_\epsilon(x)=h(x/\epsilon)$ satisfies

$$\int_Ah_\epsilon(x)dx\rightarrow0 $$ as $\epsilon$ goes to zero, for each Borel set $A$ such that $|A|<\infty$.

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    Do a change of variables.2012-03-02
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    I edited the question based on chessmath's comments on a currently deleted answer.2012-03-02
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    Could you provide an example of $f$ which satisfy the hypothesis but which is not square integrable?2012-07-26
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    @DavideGiraudo $x^{-1/2}$2012-10-03
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    What are the precise integrability assumptions on $h$? $L^1$, $L^1_\mathrm{loc}$, or what?2012-10-03
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    I guess you need at least $L^1_{loc}$. Also, does $|A|$ denote the diameter or the measure of $A$?2012-10-05

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