TITLE: Passing a limit into expectation
QUESTION [8 upvotes]: While reading about random walks, I started thinking about this and got a headache:
Given a random process $\{X_n\}_{n\in \mathbb{Z}^+}$ with a real state space (i.e., $X_n$ takes on real numbers), what can you say about these two expressions?
$$\begin{align*}
\lim_{n\rightarrow\infty} \mathbb{E}\left[X_n\mathop{\big|}X_0=j\right] & \quad (1)
\\ \mathbb{E}\left[\lim_{n\rightarrow\infty} X_n\mathop{\big|} X_0 = j\right] & \quad(2) 
\end{align*}$$
(or also considering omitting the conditional statement on $X_0$... I'm not sure whether it matters or not).
I did some Googling, and I found something called Fatou's lemma, but I have no background in measure theory, and most of the Wikipedia article is way over my head.
My questions

When can you pass the limit in and out (equality), and when is it an inequality (Fatou's lemma)?
Does anything change if $\{X_n\}$ is a Markov chain?
What if it was a continuous process $\{X_t\}_{t\in\mathbb{R}^+}$ instead?
How do you interpret (explain in words) what those two expressions mean and what the difference is between them?
Is there any causal relationship between existence of one limit and the other? For example, does the existence of the limit in $(1)$ imply that the limits in $(2)$ all exist?
Besides measure theory and probability theory, are there any other foundational topics/subjects/fields that I could look into to learn more about this?

I'm sorry if my questions don't make much sense or if I am using terminology incorrectly; I am not very familiar with this material, and would like to learn more about it. Any helpful information would be much appreciated.

REPLY [9 votes]: Try Lebesgue dominated convergence theorem: if $X_n\to X$ almost surely with respect to the probability measure $\mathbb P_j(\ )=\mathbb P(\ \mid X_0=j)$, and if $|X_n|\leqslant Y$ almost surely and for every $n$, with $\mathbb E_j(Y)$ finite, then $\lim\limits_{n\to\infty}\mathbb E_j(X_n)=\mathbb E_j(X)$, that is, $$\lim\limits_{n\to\infty}\mathbb E(X_n\mid X_0=j)=\mathbb E\left(\lim\limits_{n\to\infty}X_n\,{\Large\mid}\, X_0=j\right).$$