Almost sure convergence and convergence in mean

Question

I know that if $X_n\rightarrow Y$ almost surely, this does not necessarily imply that $\lim_{n\rightarrow\infty}\mathbb{E}|X_n-Y|\rightarrow 0$ (this would require additional proof, using for example uniform integrability). However, I am interested in a weaker statement: $$\lim\mathbb{E}X_n-\mathbb{E}Y\rightarrow 0.$$ Can this property be deduced from almost surely convergence?