The limit of the integral when the set is decreasing in probability to zero. This is an exercise problem(#2 in section 3.2) from 'A course in probability theory'.
If $E(\vert X \vert ) < \infty$ and $\lim_{n \to \infty} P(A_n) = 0,$ then $\lim_{n \to \infty} \int_{A_n} X \ dP = 0.$
I tried using the modulus inequality of integrals but I cannot get the result. Thank you.

Since $X1_{A_n}\xrightarrow{p}0$ and $|X1_{A_n}|\le |X|$ with $\mathbb{E}|X|<\infty$ the dominated convergence theorem (for sequences converging in prob.) implies that $\mathbb{E}X1_{A_n}\rightarrow 0$.
