I am not very familiar with the measure theoretic approach to probability which might be why I cannot answer the following question to myself.
Say we have a discrete and finite random variable $X$ that can take $n$ values. Let $\mathcal{X}=\{1,...,n\}$ be the set of possible values that $X$ can take. Then the set $\mathcal{P}_{\mathcal{X}}$ of possible probability mass functions is the $n-1$-simplex. I.e.\ the set of functions $p:\mathcal{X} \rightarrow [0,1]$ such that $\sum_{x\in \mathcal{X}} p(x) =1$.
Now let us define a function $m:\mathcal{P}_{\mathcal{X}}\rightarrow \mathbb{R}$ that takes as an argument a probability mass function and maps it to a real number. My question is, is the expected value $\mathbb{E}[X]$ such a function?
What confuses me is that the expected value
$$\mathbb{E}[X]=\sum_{x \in \mathcal{X}} x \; p(x)$$
somehow needs access to the values that $X$ can take but it shouldn't be able to tell that by just looking at the function $p$. If we swap $\mathcal{X}$ to another set of natural numbers, e.g. $\{10,...,10+n\}$ then the expectation value changes but the set of probability mass functions remains exactly the same.
EDIT: As Batman noted in a comment, this is a bit imprecise. Let $\mathcal{Y} = \{10,...,10+n\}$ then we don't actually have $\mathcal{X}=\mathcal{Y}$ of course, we only have that there is a bijection (they are isomorphic). Then I suspect that we also don't have $$\mathcal{P}_{\mathcal{X}}=\mathcal{P}_{\mathcal{Y}}$$ but again only isomorphism. This then suggests that $m:\mathcal{P}_{\mathcal{X}} \rightarrow \mathbb{R}$ actually "sees" or "knows" $\mathcal{X}$ and the expectation value can be defined as such a function.
I now wonder how a mathematician would go about defining a function $g:\mathcal{P}_{\mathcal{X}} \rightarrow \mathbb{R}$ that doesn't "see" $\mathcal{X}$? I.e. a function that is automatically also defined for elements of $\mathcal{P}_{\mathcal{Y}}$? END EDIT.
The previous end to this question might still be relevant: This seems to suggest that the expectation value is not a function of the probability mass function. It should be a function of the random variable in some sense however.
Does this mean that the expectation value is a function on
$$\mathcal{P}_{\mathcal{X}} \times \{\mathcal{A}\subset \mathbb{Z}:|\mathcal{A}|=n\}$$
i.e. on the pairs of elements of the $n-1$-simplex and subsets of the integers of cardinality $n$?
If that is so, what is the accurate mathematical terminology for this? Do some people really define the expectation value like this?