Atomless Probability Measure on Euclidean Space

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, let $q\in\mathbb{N}$ satisfy $1\le q$ and let $\mu$ belong to the set $\mathcal{P}(\mathbb{R}^{q})$ of probability measures on $\mathbb{R}^{q}$. For $x\in\mathbb{R}^{q}$ the one-point set $\{x\}$ belongs to $\mathcal{B}(\mathbb{R}^{q})$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets §finite-sets, so $\mu(\{x\})$ is defined.

(Atomless measure)¶ The measure $\mu$ is atomless if