The other thing that you can't say either is that it's not deterministic, because without determinism the hash function is not useful.
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Yeah yeah, in this we agree. I was just lazy.
One should say:
X is a random variable representing the choice of a message at random (under probability distribution PX).
Then h(X) is a random variable, that induce a different probably distribution bla bla.
it's super magical stuff anyway. non-crypto people don't get it.