# Riemann-Stieltjes, Lebesgue-Stieltjes integration

Just wondering - what are the essential features of Riemann-Stieltjes and Lebesgue-Stieltjes integration, and how do they differ from the usual Riemann/Lebesgue integration? In what sense are they more 'general' than the Riemann/Lebesgue integral?

The exposition of most texts in probability theory / real analysis uses Lebesgue integration - so how 'useful', or 'relevant', are the Riemann-Stieltjes and Lebesgue-Stieltjes to day-to-day integration? (as opposed to being primarily mathematical curiosities, techniques to be reserved for extremely degenerate cases)

## Answers and Replies

Ouch, I understand now. I was earlier confused between the Lebesgue and Lebesgue-Stieltjes integrals.

Can somobody tell me teh definition of Lebesgue-Stieltjes integral with measure in 2-3lines.

mathman
Science Advisor
Can somobody tell me teh definition of Lebesgue-Stieltjes integral with measure in 2-3lines.
The following is a description of the simplest case - one dimensional integral over the real line.
Lebesgue integral is an integral developed using measure theory. Measure theory starts with the idea of measurable sets. For ordinary Lebesgue integral, the measure of an interval is its length.
Lebesgue-Stieltjes integral is an integral using a measure, where the measure of an interval is not necessarily the length of the interval.