Let $\mu$ be a probability measure on $X$, so that $\int_X \mu(dx) = 1$.
Say under which conditions on the function $f: X \rightarrow \mathbb{R}_{> 0} \ $ (that is measurable and integrable) we have that
$$ \lim_{\mu(A) \rightarrow 0 } \int_A f(x) \mu(dx) = 0 $$