Diffuse Measure of Countable Set

Theorem
Let $\left({X, \mathcal A, \mu}\right)$ be a measure space.

Suppose that for all $x \in X$, the singleton $\left\{{x}\right\}$ is in $\mathcal A$.

Suppose further that $\mu$ is a diffuse measure.

Let $A \in \mathcal A$ be a countable measurable set.

Then $\mu \left({A}\right) = 0$.