This is a question coming from a discussion between Ilya and me (Thanks, Ilya!):
a random variable standalone is exactly a representation of a distribution, nothing more.
However, when it comes to several random variables, they allow operating with distributions in a more "natural" way e.g. you can sum them etc, which would correspond to the convolution of distributions which is less easy to catch.
So I was wondering if there are cases (or applications) when random variables cannot be identified with their distributions, but can only be treated as measurable mapping? In other words, can everything about random variables be completely rephrased in terms of their distributions alone, without referring to them as measurable mappings?
One particular example I am now thinking is whether everything about stochastic processes can be rephrased in terms of their laws completely?
Thanks and regards!