Almost sure convergence to infinity, does mean also converge?

Question

Consider a sequence of non-negative random variables $X_N \geq 0$ such that $X_N \rightarrow X$ a.s. where $X$ has the property that $E[X] = +\infty$. Can we then say that $\lim_{N \rightarrow \infty} E[X_N] = + \infty$? Is there a counter-example? What if instead of the condition $E[X] = + \infty$ we had $P[X = +\infty] > 0$?