I have two easy question but that I'd like to be sure of the answer:
$1)$ when you are looking at the supremum of a r.v. so defined: $$\sup_{t\in \mathbb R}X_t$$ the supremum is over the real number excluded infinity, right?
$2)$ Can we apply the monotone convergence theorem, or Lebesgue's theorem, or Fatou's Lemma when we have the following? $$\lim_{n \to \infty}E[|X_n|]\to E[|X|]$$ I should check that $|X_n|$ (Notice: on the absolute value of the r.v. right?) satisfy the condition of the relative theorem and then I can apply it, right?