Can there be metrics on sets of random variables?

Question

First off - I do not know much probability theory, so please pardon me if this question is nonsensical.

The question arose from the following thought: can I make the expectation function continuous, by assigning to a set of random variables (say, those with finite expectation), some sort of metric (or even, just a topology)?

Would be interested to hear about this, if the question makes sense.

Answer 1

The obvious example is the $L^1$ norm: $$\|X\|=E(|X|).$$ For different notions of convergence of random variables see... Convergence of random variables.