Translation-Invariant Measures on ℝⁿ Are Multiples of Lebesgue Measure

In the setting of Product Measures, Lebesgue Measure on ℝⁿ and Image Measures: Notation, assume the axiom of dependent choice. Let $n\in\mathbb{N}$ and let $\mathbf{u}=(1,\dots,1)\in\mathbb{R}^{n}$, so that $R(0,\mathbf{u})$ is the open unit cube.

(i) Multiples of Lebesgue measure. Let $\mu$ be a measure on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ such that

$$\mu(B+a)=\mu(B)\qquad(B\in\mathcal{B}(\mathbb{R}^{n}),\ a\in\mathbb{R}^{n})$$

and $c=\mu(R(0,\mathbf{u}))<\infty$. Then

$$\mu(B)=c\,\lambda_{n}(B)\qquad(B\in\mathcal{B}(\mathbb{R}^{n})),$$

the product in $[0,\infty]$.