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

Let $n\ge1$ be a natural number, let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$-algebra $\mathcal{B}(\mathbb{R}^n)$ of Euclidean space $\mathbb{R}^n$, and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^n$ whose open sets form a topology by Metric Open Sets Form a Topology. Metric and topological notions below refer to $d$ and that topology.

1. (Balls) For every $x_0\in\mathbb{R}^n$ and every real number $r>0$, the open ball of centre $x_0$ and radius $r$ is a Borel set of positive $\lambda_n$-measure.

2. (Bounded sets) Every bounded set $B\in\mathcal{B}(\mathbb{R}^n)$ satisfies $\lambda_n(B)<\infty$.

3. (Compact sets) Every compact subset $K$ of $\mathbb{R}^n$ is a Borel set with $\lambda_n(K)<\infty$.