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I need a little help understanding how to compute a Lebesgue-Stieltjes integral. If $\{s_n\}_n$ is a sequence of real numbers such that $s_n>0$ $\forall n$ and the sum of all the $s_n$ isn't infinite, you can define a measure $\mu(B) = \sum_{n \in B}s_n$, where $B$ is a Borel set. This has a distribution function $F(x)=\sum_{n=1}^{[x]}s_n$.

So, lets say there is a Borel-Measurable function $f$, and I want to compute the Lebesgue Stieltjes integral $\int_{\mathbb{R}}fdF=\int_{\mathbb{R}}fd\mu$. How do I deal with the "$d\mu$" part here? Obvious i need to take the integral with respect to my measure but I am somewhat lost and confusing myself ever further.


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Each singleton set $\{n\}$ has measure $s_n$, and open intervals $(n,n+1)$ has measure $0$ according to $\mu$, so $$\int_{\Bbb R}fd\mu=\sum_{n} f(n)\cdot s_n.$$

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