Proof of Balls Have Positive Lebesgue Measure and Bounded Sets Have Finite Lebesgue Measure

Write $\lVert\,\cdot\,\rVert$ for the Euclidean norm and $\lambda$ for Lebesgue measure on the Borel $\sigma$-algebra of the real line. By claim 2 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ we have $d(x,y)=\lVert x-y\rVert$ for all $x,y\in\mathbb{R}^n$; in particular $d(x,0)=\lVert x\rVert$, since $x-0=x$ in the real vector space $\mathbb{R}^n$. By claim 1 of Finite Products of Lebesgue Measure and Coordinate Integration on $\mathbb{R}^l$ the $\sigma$-algebra there is generated by the Borel rectangles, so every Borel rectangle belongs to it, and 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 that $\sigma$-algebra is $\mathcal{B}(\mathbb{R}^n)$.

Claim 1. Let $x_0=(x_{0,1},\dots,x_{0,n})$ and let $B(x_0,r)$ be the open ball of centre $x_0$ and radius $r$. It is open in $(\mathbb{R}^n,d)$ by Open Ball in a Metric Space is Open, hence Euclidean open by Euclidean Openness Agrees with Metric Openness on $\mathbb{R}^n$, hence Borel by claims 4 and 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.

Let $c$ be the image of $n$ under the canonical map from $\mathbb{N}$ to $\mathbb{R}$. By claim 3 of Properties of the Canonical Map from the Natural Numbers to an Ordered Field we have $0<c$, and $c^{-1}$ exists and is positive; the same claim applied to the natural number $2$ shows $0<2$, so $2c$ is positive and invertible. Put $t=r\,(2c)^{-1}$, a real number with $0<t$, and for each $j$ let

an interval with endpoints $x_{0,j}-t$ and $x_{0,j}+t$; it is a Borel subset of $\mathbb{R}$ and $\lambda(I_j)=2t$ by claim 4 of Existence of Lebesgue Measure on the Real Line. Set $Q=I_1\times\dots\times I_n$, a Borel rectangle, so that $Q\in\mathcal{B}(\mathbb{R}^n)$ and, by Lebesgue Measure on $\mathbb{R}^n$,

a product of $n$ factors each equal to the positive real number $2t$. Since the product of two positive elements of an ordered field is positive, induction on the number of factors, using Principle of Induction for the Natural Numbers, gives $0<\lambda_n(Q)$.

Now let $x=(x_1,\dots,x_n)\in Q$. For each $j$ we have $-t<x_j-x_{0,j}<t$, so $|x_j-x_{0,j}|\le t$ for the absolute value. By Coordinate Bounds Control the Euclidean Norm applied to the point $x-x_0$,

the last inequality because $0<r$. Hence $x\in B(x_0,r)$, so $Q\subseteq B(x_0,r)$, and claim 2 of Basic Properties of a Measure gives

Claim 2. Let $B\in\mathcal{B}(\mathbb{R}^n)$ be bounded, so there are a point $x_1\in\mathbb{R}^n$ and a real number $R_0>0$ with $d(x_1,y)\le R_0$ for every $y\in B$. Put $R=R_0+d(x_1,0)$, a real number with $0<R$. For $y\in B$, the symmetry and the triangle inequality of the metric $d$ give

By claim 4 of Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ this yields $|y_j|\le\lVert y\rVert\le R$ for every $j$, so $y$ lies in the Borel rectangle $J\times\dots\times J$, where $J=\{s\in\mathbb{R}:-R\le s\text{ and }s\le R\}$ is the interval with endpoints $-R$ and $R$, of measure $\lambda(J)=2R$ by claim 4 of Existence of Lebesgue Measure on the Real Line. Thus $B\subseteq J\times\dots\times J$ and, by Lebesgue Measure on $\mathbb{R}^n$, the measure of that rectangle is a product of $n$ real factors each equal to $2R$, hence a real number and in particular smaller than $\infty$. Claim 2 of Basic Properties of a Measure now gives $\lambda_n(B)\le\lambda(J)\cdots\lambda(J)<\infty$.

Claim 3. Let $K\subseteq\mathbb{R}^n$ be compact. By Compact Subset of $\mathbb{R}^n$ is Closed it is closed, hence Borel by claims 4 and 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, and by Compact Subset of $\mathbb{R}^n$ is Bounded it is bounded. Claim 2 applies and gives $\lambda_n(K)<\infty$.