Let $(\Omega, \mathcal A, P)$ a measure space and $X:\Omega→\mathbb R, Y:\Omega → \mathbb R$ two random variables. As far as I know
$$X+Y:=X(\omega)+Y(\omega) \ \forall \omega \in \Omega, $$ so just the pointwise sum, is this correct? Likewise, we should have $$XY:=X(\omega)Y(\omega) \ \forall \omega \in \Omega, $$ right?
So what is the definition of $$\max\{X,Y\}?$$ Is it just $\max_{\omega \in \Omega} \{X(\omega),Y(\omega)\}$?
Any help is appreciated!
The object $\max\{X,Y\}$ is itself a random variable. If I give it a name, say $Z=\max\{X,Y\}$, then $$ Z(\omega) = \max\{X(\omega), Y(\omega)\}. $$ So the $\max$ is still only over the two numbers, but it is performed separately for each individual $\omega$.
Your definitions of $X+Y$ and $XY$ are exactly correct. More generally, if $f(x,y)$ is any measurable function of two variables, we can create a new random variable $Z=f(X,Y)$, where $Z(\omega)=f(X(\omega),Y(\omega))$. What we are talking about here are the special cases $f(x,y)=x+y$, $f(x,y)=xy$, and $f(x,y)=\max\{x,y\}$.