Lebesgue Measure on $\mathbb{R}^n$

Let $n\ge1$ be a natural number, let $\mathcal{B}(\mathbb{R}^n)$ be the Borel $\sigma$-algebra on Euclidean space $\mathbb{R}^n$, and let $\lambda$ be Lebesgue measure on the Borel $\sigma$-algebra of the real line.

Lebesgue measure on $\mathbb{R}^n$ is the measure $\lambda_n$ on $\mathcal{B}(\mathbb{R}^n)$ satisfying

$$\lambda_n(A_1\times\dots\times A_n)=\lambda(A_1)\cdots\lambda(A_n)$$

for all Borel subsets $A_1,\dots,A_n$ of $\mathbb{R}$, the product being formed in $[0,\infty]$ with the conventions of Measure, Measure Space, and Probability Measure extended by $a\cdot\infty=\infty\cdot a=\infty$ for $0<a\le\infty$. Exactly one measure on the $\sigma$-algebra $\mathcal{B}_n$ of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ has these values, by claim 1 there, and $\mathcal{B}_n=\mathcal{B}(\mathbb{R}^n)$ by claim 5 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets; $\lambda_n$ is that measure. For $n=1$ it is $\lambda$ itself.