Dependency of the expectation

Question

I have read a few times now in some books the following sentence:

"Expectation depends only on its distribution."

I only had the case that $X$ is a real-valued random variable on a countable space Ω. What does this sentence mean and how can one show it?

Answer 1

I would rather rephrase:

The expectation of a random variable is completely determined by the distribution of that random variable.

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If $(\Omega,\mathcal A,P)$ is a measure space where $P$ is a probability measure and $X:\Omega\to\mathbb R$ is a random variable with existing expectation then by definition:$$\mathbb EX=\int X(\omega)P(d\omega)$$The random variable induces a probability measure $P_X$ on measurable space $(\mathbb R,\mathcal B)$ where $\mathcal B$ denotes the $\sigma$-algebra of Borel subsets of $\mathbb R$.

This $P_X$ is prescribed by $B\mapsto P(\{X\in B\})$ and is the distribution of $X$.

It can be shown that: $$\mathbb EX=\int xP_X(dx)$$ where the RHS is completely determined by $P_X$.

In that sense the expectation of $X$ is determined by the distribution of $X$.