Nothing we have done with random variables really requires the variables to take values in \(\mathbb{R}\).
It will often be useful to work with random vectors: \(X:E\to\mathbb{R}^n\).
Here, the random vector produces a probability measure on Borel sets in \(\mathbb{R}^n\).
Each coordinate of the vector is a random variable. The expected value of the vector is the vector of expected values.