Notation in probability: what is the domain of the expectation operator?

Question

Let $X$ be a random variable/vector, $X : \Omega \to \mathbb{R}^n$.

Let $\mathbb{E}$ be the expectation operator

What is the domain of $\mathbb{E}$?

Two possibilities: 1. set of measurable functions, or 2. $\mathbb{R}^n$

Can't be sure which one.

Answer 1

You calculate the expected value of a random variable, so the domain of the expectation operator is the set of random variables. These are measurable functions on the probability space.

The codomain is the set where the random variable takes its values. Your example concerns vector valued random variables, so the expectation will be a vector.