It seems that one usually use the set of all possible values of $X$ as the sample space of random variable $X$. (Therefore discrete random variables have countable sample space, continuous random variables have uncountable sample space.) However, I don't think this is right. Since random variables are defined as measurable functions over sample space, the above assumption would make sum/product of a discrete random variable and a continuous random variable meaningless (because they are function over different spaces.)
Well, this post says that
Every uncountable standard Borel space is isomorphic to [0,1] with the Borel σ-algebra. Moreover, every non-atomic probability measure on a standard Borel space is equivalent to Lebesgue-measure on [0,1].
However, this doesn't solve all the problems.
First, that claim is true only for uncountable sample space, so the problem of adding of a discrete RV and a continuous RV is still unsolved.
Second, even with two continuous RV, there are still problems if we consider expectation (integration). Although the quoted claim says that "every non-atomic probability measure on a standard Borel space is equivalent to Lebesgue-measure on [0,1].", however, two different measures cannot be transformed to Lebesgue measure on [0,1] with same isomorphism. Therefore two RVs $X$ and $Y$ may have different measures on [0,1], which will make $\mathbb{E}XY$ meaningless.
This really confused me.... Can people really base all different random variables onto one same sample space with one same measure?