My professor seems to randomly consider the product $XY$ or the sum $X+Y$ of random variables $X$, $Y$. Do they always exist as a random variable, too?
Upon some research I learned that the random variables with expected value $E\left[\left|X\right|^p\right]<\infty$ form a vector-space. So the sum $X+Y$ as linear combination of random variables is a random variable. But I found no explanation for the existence of the product $XY$ (also not for $|X|$).