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 $|A|<\infty$.