$\inf$ and $\sup$ of two real random variables

Question

Let $X$ and $Y$ be two rrv’s. Prove that $\inf(X,Y)(w):=\inf \{X(w),Y(w)\}$,$w\in \Omega$ and $\sup(X,Y)(w):=\sup\{X(w),Y(w)\}$, $w\in \Omega$ are also rrv’s.

In the book, definition of real random variable is given. I am confused how we’re going to prove this statement by using the definition. Also I know that the sum of random variables are random variables.

Answer 1

Hint: Observe that $\min(X,Y)> c$ if and only if $X>c$ and $Y> c$. Therefore $$ \{\min(X,Y)> c\} = \{X> c\}\cap\{Y> c\}. $$ So if $X$ and $Y$ are real random variables, then for every $c$ the sets $\{X> c\}$ and $\{Y> c\}$ belong to the sigma-algebra $\cal F$, and therefore...