Can random variables always be identified with their distributions?

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!

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The answer is clearly no in the literal sense, but terry Tao has an interesting post on what is supposed not to matter in probability theory. –  Michael Greinecker Jan 19 '13 at 17:22
@MichaelGreinecker: Thanks! Do you refer to his "probability theory is only allowed to study concepts and perform operations which are preserved with respect to extension of the underlying sample space"? Can this sentence, together with his analogy to differential geometry and graph theory, be explained from category's or some other general view? –  Tim Jan 19 '13 at 17:51
Yes, that's what I mean. I'm not sure about some generalizing point of view, I will think about it. –  Michael Greinecker Jan 19 '13 at 19:55
@MichaelGreinecker: (1) I in particular don't quite understand why the extension is defined to be surjective in Tao's blog. Is the concept "extension" trying to become morphisms in some category of probability spaces? (2) I found two links theoreticalatlas.wordpress.com/2010/11/11/… and mathoverflow.net/questions/49426/…. Although I can only understand some small part of them, I hope they can be interesting to you. –  Tim Jan 19 '13 at 20:09
The extension is define to be surjective as we define it from the extended space to the original space - hence we want to capture everything that happens on the original space (and perhaps also add something new) - that's why the map has to be surjective. Otherwise, we are just extending the range of the map. –  Ilya Jan 20 '13 at 10:21