Riemann-Stieltjes, Lebesgue-Stieltjes integration

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In summary: It is more general than the usual Riemann/Lebesgue integral, and is often used in probability theory and real analysis. Its main application is in defining the cumulative distribution function of a random variable. It differs from the Riemann/Lebesgue integral in that it uses a measure instead of just the length of an interval. It is useful in many practical situations and not just limited to degenerate cases.
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guiness
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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)
 
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Ouch, I understand now. I was earlier confused between the Lebesgue and Lebesgue-Stieltjes integrals.
 
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Can somobody tell me teh definition of Lebesgue-Stieltjes integral with measure in 2-3lines.
 
  • #5
tushi said:
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.
 

1. What is the difference between Riemann-Stieltjes and Lebesgue-Stieltjes integration?

Riemann-Stieltjes integration is a method of integration that involves integrating with respect to a function known as the Stieltjes function. This method is based on partitioning the interval into smaller subintervals and approximating the area under the curve using rectangles. Lebesgue-Stieltjes integration, on the other hand, is a more general form of integration that is based on the concept of measure theory. It allows for integration with respect to a wider class of functions, including non-monotonic and discontinuous functions.

2. What are some applications of Riemann-Stieltjes and Lebesgue-Stieltjes integration?

Riemann-Stieltjes and Lebesgue-Stieltjes integration are used in various fields such as physics, engineering, economics, and probability theory. They are particularly useful in solving problems involving the calculation of areas, volumes, and moments of a function. They are also used in the study of differential equations and in the development of statistical models.

3. Can you explain the concept of a Stieltjes measure?

A Stieltjes measure is a type of measure that is used in Lebesgue-Stieltjes integration. It is defined as a function that assigns a non-negative value to each interval of the real line. This measure is used to extend the concept of integration to a wider class of functions, allowing for the integration of non-monotonic and discontinuous functions.

4. How is Lebesgue-Stieltjes integration related to Lebesgue integration?

Lebesgue-Stieltjes integration is a generalization of Lebesgue integration. Both methods are based on the concept of measure theory and allow for integration with respect to a wider class of functions. However, Lebesgue-Stieltjes integration includes the added component of a Stieltjes measure, which allows for the integration of a larger class of functions.

5. Are there any drawbacks to using Lebesgue-Stieltjes integration?

One potential drawback of Lebesgue-Stieltjes integration is that it can be more complicated and difficult to use compared to other methods of integration, such as Riemann integration. It also requires a solid understanding of measure theory, which can be challenging for some individuals. However, the benefits of this method, such as its ability to integrate a wider class of functions, often outweigh these drawbacks.

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