Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit as $i \rightarrow \infty$, $$ \mathbb{E}(X_i) = 0 $$ where $\mathbb{E}$ denotes expectation?
On the otherhand let $Y_1, Y_2, Y_3 \dots$ be a sequence of random variables and in addition we have, $$ \mathbb{E}(Y_i) = 0 $$ does it follow that in the limit as $i \rightarrow \infty$, $$ Y_i \rightarrow 0 \text{ almost surely} $$ I would love to see a proof or counterexample.
The expectation of $X_i$ does not necessarily exist, but even if it does, the sequence of expectations may not converge to $0$. For example consider a probability measure $\mathbb P$ on the set of positive integers such that $\mathbb P\{n\}= 2^{-n}$ (here $\Omega=\mathbb N\setminus\{ 0\}$). Define $X_i$ as $2^i\cdot\mathbb 1_{\{i\} } $, that is, $X_i(\omega)=2^i$ if $\omega=i$ and $0$ otherwise. Then $X_i\to 0$ almost surely but $\mathbb E[X_i]=1$.
We can modify the example in order to have $\mathbb E[X_i]=a$ where $a $ is a fixed number and we may even have $$\mathbb E[X_i]\to \infty$.
Take $Y_i=1$ with probability $1/2$ and $-1$ with the same probability.