The Lebesgue Measure of a Closed Ball in $\mathbb{R}^n$

We work in the setting of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background, whose notation is fixed for every dimension and is used here with a natural number $n$ satisfying $1\le n$: the Euclidean norm $\lVert\,\cdot\,\rVert$, distance $d_{E}$, topology, the notions of open, closed, bounded and compact subsets, and the open and closed balls $B(x,r)$ and $\bar{B}(x,r)$ of $\mathbb{R}^{n}$, the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^{n})$, and Lebesgue measure $\lambda_{n}$ with its conventions for $[0,\infty]$ are as fixed there. We write $0$ for the zero vector of the real vector space $\mathbb{R}^{n}$, and for $c\in\mathbb{R}$ and $B\subseteq\mathbb{R}^{n}$ we write $cB=\{cz:z\in B\}$ and $B+x=\{z+x:z\in B\}$, as in Scaling Lebesgue Measure on ℝⁿ by a Diagonal Matrix and Translation and Reflection Invariance of Lebesgue Measure on ℝⁿ. Then the following hold.

1. (Closed balls are Borel and of finite measure) (i) For every $x\in\mathbb{R}^{n}$ and every $r\in\mathbb{R}$ with $0<r$ one has $\bar{B}(x,r)\in\mathcal{B}(\mathbb{R}^{n})$ and $\lambda_{n}(\bar{B}(x,r))<\infty$.

2. (The normalising constant) (ii) By claim 1 the number $\kappa_{n}=\lambda_{n}(\bar{B}(0,1))$ is a real number, and it satisfies $0<\kappa_{n}<\infty$.

3. (The measure of a closed ball) (iii) For every $x\in\mathbb{R}^{n}$ and every $r\in\mathbb{R}$ with $0<r$ one has

$$\lambda_{n}\bigl(\bar{B}(x,r)\bigr)=\kappa_{n}\,r^{n};$$

in particular the measure of a closed ball does not depend on its centre, and it is positive.