Lebesgue Measure of Scalar Multiple

Theorem
Let $\lambda^n$ be the $n$-dimensional Lebesgue measure on $\R^n$ equipped with the Borel $\sigma$-algebra $\mathcal B \left({\R^n}\right)$.

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

Then $\lambda^n \left({t \cdot B}\right) = t^n \lambda^n \left({B}\right)$, where $t \cdot B$ is the set $\left\{{t \mathbf b: \mathbf b \in B}\right\}$.

Proof
It follows from Rescaling is Linear Transformation that the mapping $\mathbf x \mapsto t \mathbf x$ is a linear transformation.

Denote $t \, \mathbf{I}_n$ for the matrix associated to this linear transformation by Linear Transformation as Matrix Product.

From Determinant of Rescaling Matrix, it follows that:


 * $\det \left({t \, \mathbf{I}_n}\right) = t^n$

From Inverse of Rescaling Matrix, $t \, \mathbf{I}_n$ is the inverse of $t^{-1} \mathbf{I}_n$.

Thus, it follows that:

Now recall $\det \left({t^{-1} \, \mathbf{I}_n }\right)^{-1} = \det \left({t \, \mathbf{I}_n}\right) = t^n$.

Since $t > 0$, $\left\vert{t^n}\right\vert = t^n$, and the result follows.