Does almost sure convergence implies convergence of the mean? I asked a slightly similar question here: Does Convergence in probability implies convergence of the mean?, but now I wish to examine a stricter scenario:
Let $\{X_n\}_{n=1}^\infty$ be a sequence of random variables converging a.s to a const $c$. Is it required for the sequence to be uniformly integrable in order to imply  $\lim_{n\to \infty} EX_n = c$? 
And what about $\lim_{n\to \infty} EYh(X_n) = E[Y]h(c)$ for some random variable $Y$ and a continuous function $h$?  Under which regularity conditions does the last equality holds?

Think again here $~X_n~$ not converges to $~0~$ almost surely. When you are taking $~X_n=2^n~$ then probability approaches towards $~0~$ but not zero.
Definition is $~P[\lim X_n=0]=1~$, but here when $~X_n=0~$ probability is not exactly $~1~$, lies near $~1~$; hence contradicts the definition.
