Theorem

random vectors are tuples of random variables

$\mathbf{X} = (X_1 , \dots , X_n )$ is a random vector $\mathbf{X} \in (\Omega, \mathcal{B})/(\mathbb{R}^{n}, \mathcal{B}(\mathbb{R}^{n}))$ if and only if for each $i = 1, \dots , n$ it is the case that $X_i$ is a random variable.