Mapping between Euclidean Spaces Measurable iff Components Measurable

Theorem
Let $\R^n$ and $\R^m$ be Euclidean spaces.

Denote by $\mathcal{B}^n$ and $\mathcal{B}^m$ their respective Borel $\sigma$-algebras.

Denote with $\mathcal B$ the Borel $\sigma$-algebra on $\R$.

Let $f: \R^n \to \R^m$ be a mapping, and write:


 * $f \left({\mathbf x}\right) = \begin{bmatrix}f_1 \left({\mathbf x}\right) \\ \vdots \\ f_m \left({\mathbf x}\right)\end{bmatrix}$

with, for $1 \le i \le m$, $f_i: \R^n \to \R$.

Then $f$ is $\mathcal{B}^n \, / \, \mathcal{B}^m$-measurable iff:


 * $\forall i:f_i: \R^n \to \R$ is $\mathcal{B}^n \, / \, \mathcal B$-measurable