Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(X_{n}\) has distribution \(F_{n}\) for all \(n \in \mathbb{N}\). Suppose that \(X_{n} \xrightarrow{\text { a.c. }} X\) as \(n ightarrow \infty\) for some random variable \(X\). Suppose that there exists a subset \(A \subset \mathbb{R}\) such that

\[\int_{A} d F_{n}(t)=1\]

for all \(n \in \mathbb{N}\), and

\[\int_{A} d t<\infty\]

Do these conditions imply that \(E\left(X_{n}ight) ightarrow E(X)\) as \(n ightarrow \infty\) ?