A well-known result is that we can always construct a countably additive function $\mu$ from a nondecreasing and right-continuous function $G$. More specifically, we define on the semiring $\mathcal{C}$ of all intervals $(a,b]$, $\mu((a,b])=G(b)-G(a),$ where $\mu$ is the Lebesgue measure when $G$ is the identity mapping. I'm curious if the following function gets this property as well:
Fix a countable set $C=\{c_n:n\in\mathbb{N}\}\subset\mathbb{R}$ where each $c_n$ is distinct, and a countable set $\{a_n:n\in\mathbb{N}\}\subset\mathbb{R}$ where each $a_n$ is non-negative and $\sum_na_n<\infty$.
Define $G:\mathbb{R}\rightarrow\mathbb{R}$ such that $x\longmapsto\sum\{a_n:c_n\leq x\}.$
That is, is this function right-continuous and non-decreasing? Furthermore, does the measure $\mu$ constructed from this function satisfy $\mu(\{c_n\})=a_n$ and $\mu(\mathbb{R}\backslash C)=0?$
EDIT: $G$ being right-continuous and non-decreasing seems pretty easy to see. Non-decreasing is rather obvious from the property of the $a_n$'s and I basically argued right-continuity to myself in the comment box below.
However for the second part, I'm not sure how to approach either property. How do you interpret $\mu(\{c_n\})$ to get only $a_n$ remaining in the subtraction? I appreciate any help!