Constructing a random vector from random variables

Question

Let $X$ and $Y$ be random variables on a probability space $(\Omega,\mathcal{F},P)$. Consider the function $$Z:\Omega\to\mathbb{R}^2,\qquad \omega\mapsto(X(\omega),Y(\omega)).$$ If $B$ is a Borel set in $\mathbb{R}^2$, then is it true that $Z^{-1}(B)\in\mathcal{F}$ ? That is, is $Z$ a random vector? If so, how could I prove that?

Answer 1

If $B=B_1\times B_2$ where $B_1,B_2$ are Borel sets in $\mathbb R$ then $Z^{-1}(B)=X^{-1}(B_1) \cap Y^{-1}(B_2) \in \mathcal F$. Now consider the collection of all Borel sets $B$ in $\mathbb R^{2}$ for which $Z^{-1}(B) \in \mathcal F$.Verify that this is sigma algebra . Since contains all sets of the form $B_1\times B_2$ where $B_1,B_2$ are Borel sets in $\mathbb R$ it contains all Borel sets in $\mathbb R^{2}$, as required.