Euclidean Borel Sigma-Algebra Closed under Scalar Multiplication

Theorem
Let $\mathcal B \left({\R^n}\right)$ be the Borel $\sigma$-algebra on $\R^n$.

Let $B \in \mathcal B$, and let $t \in \R_{>0}$.

Then also $t \cdot B := \left\{{t \mathbf b: \mathbf b \in \mathcal B}\right\} \in \mathcal B$.