{$X_n$} be a sequence of r.v. and N is a $\mathbb{N}$-valued r.v. Prove $X_N$ is also a r.v.

Question

Let {$X_n$} be a sequence of random variables for $n \in \mathbb{N}$, and $N$ be a random variable with a value of positive integer.

{$X_n$} and $N$ are defined as on the same probability space ($\Omega, F, P$), and I would like to prove that $X_N$ is also a random variable.

How should I define mapping on $\Omega$ to start the proof? What is the next process I need to consider as well to get the right conclusion? Does it suffice to prove $X_N$ is $F$-measurable?

Answer 1

To answer your first question, $X_N$ is the mapping that takes $\omega\in\Omega$ to the number $X_{N(\omega)}(\omega)$.