Theorem

image of random vector under measurable map

If $\mathbf{X} = (X_1, \dots , X_n)$ is a random vector and $g: \mathbb{R}^{n} \to \mathbb{R}$ satisfies $g \in \mathcal{B}(\mathbb{R}^{n})/\mathcal{B}(\mathbb{R})$ then $g(\mathbf{X})$ is a random variable. In particular, if $g$ is continuous then $g(\mathbf{X})$ is a random variable.