This is from exercise 4.4 of Elstrodt's measure theory textbook. By a jump function $F:\mathbb{R}\rightarrow\mathbb{R}$ we mean a function which can be written in the form $$F(x)= \begin{cases}\phantom{-}\sum_{y\in A\cap(0,x]}p(y)&\text{if $x\geq0$}\\-\sum_{y\in A\cap(x,0]}p(y)&\text{if $x<0$}\end{cases}$$ where $A$ is some subset of $\mathbb{R}$ and $p:A\rightarrow(0,\infty)$ is a function satisfying the 'local summability' condition; that is, we require that $\sum_{y\in A\cap B}p(y)<\infty$ for every bounded subset $B$ of $\mathbb{R}$. (This makes $A$ necessarily a countable subset.)
We now consider the outer measure induced by $F$. Writing this by $\eta_F$, we have, by definition, $$\eta_F(S)=\inf\left\{\sum_{i=1}^{\infty}(F(b_i)-F(a_i)):S\subset\bigcup_{i=1}^{\infty}(a_i,b_i]\right\}$$ where $S$ is any subset of $\mathbb{R}$ and $a_i,b_i\in\mathbb{R}$. With this outer measure, the Caratheodory construction now gives the $\sigma$-algebra of $\eta_F$-measurable subsets. The problem is to show that EVERY subset of $\mathbb{R}$ is $\eta_F$-measurable. I've added the original German text below for clarity.
What I have tried: The problem is equivalent to showing that $\eta_F(S\cup T)=\eta_F(S)+\eta_F(T)$ for every disjoint subsets $S,T$ of $\mathbb{R}$. I figured that maybe $\eta_F(S)=\sum_{y\in A\cap S}p(y)$ holds for every $S$, and tried to prove this. I've succeeded in showing $\eta_F(S)\geq\sum_{y\in A\cap S}p(y)$, but got stuck on the other inequality. Right now I'm not quite sure whether this is the right idea... Any advice is welcome!