Translation Invariant Measure on Euclidean Space is Multiple of Lebesgue Measure

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

Suppose that $\mu$ is translation-invariant.

Also, suppose that $\kappa := \mu \left({\left({0 .. 1}\right)^n }\right) < +\infty$.

Then $\mu = \kappa \lambda^n$, where $\lambda^n$ is the $n$-dimensional Lebesgue measure.