Say I have two random variables $X$ and $Y$ defined on a probability space ($\Omega,\mathbb{F},\mathbb{P}$).
To prove $X+Y$ is also a random variable, I need $\{\omega:(X+Y)(\omega)\le x\}\in \mathbb{F},\ \forall x\in \mathbb{R}.$
For positive random variables, I can simply write this set as an intersection of two sets, $\{\omega:X(\omega)\le x\}$ and $\{\omega:Y(\omega)\le x\}$, and invoke the property of the sigma algebra.
Now, how do I extend this to arbitrary-valued random variables? Or is there an intersection that is irrespective of the value of the RVs?