Existence and Uniqueness of Lebesgue Measure on ℝⁿ

In the setting of Measurable Functions and the Lebesgue Integral: Notation, assume the axiom of dependent choice. Let $n\in\mathbb{N}$, let $R(a,b)$, for $a,b\in\mathbb{R}^{n}$, be the open boxes of $\mathbb{R}^{n}$, and let $\mathcal{R}_{n}$ be the set of open boxes.

(i) Boxes generate the Borel sets. $\mathcal{R}_{n}$ is a $\pi$-system on $\mathbb{R}^{n}$ with

$$\sigma(\mathcal{R}_{n})=\mathcal{B}(\mathbb{R}^{n}).$$

(ii) Cubes cover $\mathbb{R}^{n}$. For $k\in\mathbb{N}$ let $Q_{k}=R(-k\mathbf{u},k\mathbf{u})$, where $\mathbf{u}=(1,\dots,1)\in\mathbb{R}^{n}$ is the point with every component $1$. Then

$$\mathbb{R}^{n}=\bigcup_{k\in\mathbb{N}}Q_{k}.$$

(iii) Existence and uniqueness. There is exactly one measure $\kappa$ on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ such that

$$\kappa\bigl(R(a,b)\bigr)=\prod_{i=1}^{n}(b_{i}-a_{i})$$

for all $a,b\in\mathbb{R}^{n}$ with $a_{i}<b_{i}$ for every $i\in[n]$.

(iv) Boxes determine measures. If $\mu$ and $\nu$ are measures on $(\mathbb{R}^{n},\mathcal{B}(\mathbb{R}^{n}))$ with $\mu(R)=\nu(R)<\infty$ for every $R\in\mathcal{R}_{n}$, then $\mu=\nu$.