Countable Sets are Null for an Atomless Measure, One-Point Sets are Lebesgue Null, and an Absolutely Continuous Measure is Atomless

In the setting of Probability Measures on Euclidean Space and Random Vectors: Standing Notation, whose notation of Product Measures and Lebesgue Measure on ℝⁿ: Standing Notation and Background is in force, let $q\in\mathbb{N}$ satisfy $1\le q$, let $\lambda_{q}$ be the Lebesgue measure on $\mathcal{B}(\mathbb{R}^{q})$, and let $\mu\in\mathcal{P}(\mathbb{R}^{q})$.

1. (Countable sets are null for an atomless measure)(i) Suppose that $\mu$ is atomless and let $A\subseteq\mathbb{R}^{q}$ be countable. Then $A\in\mathcal{B}(\mathbb{R}^{q})$ and $\mu(A)=0$.

2. (One-point sets are Lebesgue null)(ii) For every $x\in\mathbb{R}^{q}$ one has $\lambda_{q}(\{x\})=0$. Consequently every countable subset $A$ of $\mathbb{R}^{q}$ belongs to $\mathcal{B}(\mathbb{R}^{q})$ and satisfies $\lambda_{q}(A)=0$.

3. (Absolutely continuous implies atomless)(iii) If $\mu$ is absolutely continuous, then $\mu$ is atomless.