Let $(\Omega, \mathcal{F}, P)$ be a probability space, and let $(X_n)$, $(Y_n)$ be two sequences of nonnegative, integrable random variables. Suppose $X_n \leq 1$, $X_n \to 0$, and $Y_n \to Y$ almost surely. Then, $X_nY_n \to 0$ almost surely.
I am wondering when $\int X_nY_n dP \to 0$.
If $(Y_n)$ is uniformly integrable, then so is $(X_nY_n)$, in which case $\int X_nY_n dP \to 0$. So my question becomes
What's a condition on $(Y_n)$ that's weaker than the uniform integrability and guarantees $\int X_nY_n dP \to 0$?
As Davide Giraudo points out below, if $(X_nY_n)$ is uniformly integrable then $\int X_nY_ndP \to 0$, so yet another way to ask my question is
What's a condition on $(Y_n)$ that's weaker than uniform integrability and implies that $(X_nY_n)$ is uniformly integrable?