Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

Thanks!

share|improve this question
add comment

1 Answer 1

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.