In the lecture, we often use monotone and dominated convergence. Since I have not studied maths, I have some problems understanding it, so it would be very helpful if you could try to explain it to me in easy words.
We often prove that $\mathbb{E}[|S_k|] < \infty$, using the monotone convergence theorem and then state that this is enough to apply dominated convergence theorem on $\mathbb{E}[S_k]$. Is this always true or do we need some more assumptions?
Also, we have proved that $\mathbb{E}[X^r] < \infty$ for all $r$ (all moments of a random variable are finite) and stated that this is enough to apply dominated convergence theorem. Can you explain to me why?
Sometimes we say that if $X_Y \geq 0$, then $\mathbb{E}[X_Y \mid Y]$ is well-defined. What does this mean and why does this hold?
Moreover, I am not sure about the relation between $\mathbb{E}[X] < \infty$, $\mathbb{E}[X^2] < \infty$. Does one imply the other? Why? Where is it useful to have that these expectations are finite?
Sorry for the probably very stupid questions, but I am really confused at the moment, and thank you for your help.