Exchanging expectation and limits on a finite probability space Suppose I have an infinite sequence of random variables $X_n \rightarrow X $ a.s. 

Suppose that the sample space $\Omega$ is finite.

Does it follow that E[$\textrm{lim}_{n\rightarrow\infty} X_n] = \textrm{lim}_{n\rightarrow\infty} E[X_n]$ ?

It seems to me that it should follow, since given that $\Omega$ is finite, there should be a dominating r.v. and thus the dominated convergence theorem should apply. Is this correct?

(Apologies if this is a dumb question, I am not a mathematician by training.)

Thanks!
case your sum is diverging :    

a simple counter exmple :
let $\Omega$ = {-1;1} and define $(\epsilon_i)_i$ an infinite sequence of rademacher variables i.e. $\mathbb{P}(\epsilon_i=-1)=1/2$ then :
$\sum_{i=1}^{\infty}\mathbb{E}[\epsilon_i]=0$
But we know that $S_\infty=\sum_{i=1}^{\infty}\epsilon_i$ is divergent so it can't have a finite expectation   

Indeed if $\mathbb{E}[\sum_{i=1}^{\infty}\epsilon_i]&lt;+\infty$ then it would imply imediatly that $\sum_{i=1}^{\infty}\epsilon_i &lt; \infty$ a.s wich is known to be false (infinity will be visited i.o. it's a known proprety of the random walk, for a little more intuition on this see : https://mathoverflow.net/questions/60417/will-a-random-walk-on-0-inf-tend-to-infinity)    

case your sum is converging :

Here you have two options either your sum is made of positiv random variables and then Fubini-Tonelli applies without problem whatever the cardianality of $\Omega$
But if your sum is not made of positiv r.v. then it must satisfy absolute convergence to apply Fubini-Lebesgue theorem and so it still does not depend on the cardinality of $\Omega$ 

https://en.wikipedia.org/wiki/Fubini%27s_theorem
case your sum is diverging :    

a simple counter exmple :
let $\Omega$ = {-1;1} and define $(\epsilon_i)_i$ an infinite sequence of rademacher variables i.e. $\mathbb{P}(\epsilon_i=-1)=1/2$ then :
$\sum_{i=1}^{\infty}\mathbb{E}[\epsilon_i]=0$
But we know that $S_\infty=\sum_{i=1}^{\infty}\epsilon_i$ is divergent so it can't have a finite expectation   

Indeed if $\mathbb{E}[\sum_{i=1}^{\infty}\epsilon_i]&lt;+\infty$ then it would imply imediatly that $\sum_{i=1}^{\infty}\epsilon_i &lt; \infty$ a.s wich is known to be false (infinity will be visited i.o. it's a known proprety of the random walk, for a little more intuition on this see : https://mathoverflow.net/questions/60417/will-a-random-walk-on-0-inf-tend-to-infinity)    

case your sum is converging :

Here you have two options either your sum is made of positiv random variables and then Fubini-Tonelli applies without problem whatever the cardianality of $\Omega$
But if your sum is not made of positiv r.v. then it must satisfy absolute convergence to apply Fubini-Lebesgue theorem and so it still does not depend on the cardinality of $\Omega$ 

https://en.wikipedia.org/wiki/Fubini%27s_theorem