## Measurable Spaces – Problem (49/365)

Let $\mu$ be the Lebesque-Stieltjes measure generated by a continuous function. Show that if the set $A$ is at most countable, then $\mu(A) = 0$. A Lebesque-Stieltjes measure is a countably additive measure and is given by a generalized distribution function $G(x)$ such that $\mu([a,b)) = G(b) - G(a)$ that is continuous on the right. So,

$\displaystyle A = \cup_{i=1}^\infty \{ a_i \} \\ \mu(A) = \sum_{i=1}^\infty \mu(\{ a_i \}) \text{ by countable additivity}\\ \mu(\{ a_i \}) = \lim_{n \rightarrow \infty} \mu [a_i, a_i + \frac{1}{n}) \\ = \lim_{n \rightarrow \infty} G(a_i + \frac{1}{n}) - G(a_i) \\ = 0 \\ \text{Thus } \mu(A) = 0$