Non Stieltjes integrable function

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In summary, the function f: [-2,2] to R given by f(x) = 1 if x is greater than or equal to 0 and 0 if x is less than 0 is not Riemann-Stieltjes integrable with itself due to their shared point of discontinuity at 0. This can be justified by considering the limit of the superior and inferior sums, which do not equal for any partition. Even though the integral may seem to be equal to 1, this is not the case as the contribution of an interval like [-0.1, 0] involves more than just f(0). A proof for the theorem that functions with a common point of discontinuity are not Riem
  • #1
Damidami
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I know that this function [itex] f : [-2,2] \to \mathbb{R} [/itex]
[itex] f(x) = \begin{cases} 1 & \textrm{ if } x \geq 0 \\ 0 & \textrm{ if } x < 0 \end{cases} [/itex]
is not Riemann-Stieltjes integrable with itself (that is, taking [itex] g = f [/itex] then [itex] f \not\in R(g) [/itex])
That is because both share a point of discontinuity, namely 0.

What I don't know is how do I justify this? for which partition does the limit of the superior sums and the inferior sums don't equal? Shouldn't this integral be equal to 1? (all the terms in the sum cancel except at zero, where [itex] \triangle g = 1-0 = 1 [/itex], and [itex] f(0) = 1[/itex]

I think I'm not seeing it correctly.
 
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  • #2
Damidami said:
[itex] f(0) = 1[/itex]

In computing the upper and lower sums, the contribution of an interval like [-0.1, 0] involves more than f(0). You have to use the infimum of f on the interval when computing the lower sum.

I found a PDF of class notes that supposedly proves the theorm that if the two functions share a common point of discontinuity then they are not Riemann-Stieljes integrable. (page 3 of 5) It looks like it's hard to prove!

http://www.google.com/url?sa=t&rct=...sg=AFQjCNEL7sm8dOrwngWsUyk-yVKajXGqQA&cad=rja
 

What is a Non Stieltjes integrable function?

A Non Stieltjes integrable function is a type of function that cannot be integrated using the Stieltjes integral, which is a generalization of the Riemann integral. It is defined as the limit of a Riemann sum, where the function is multiplied by a weight function. Non Stieltjes integrable functions have certain properties that make them different from Stieltjes integrable functions, such as not being continuous or bounded.

What are some examples of Non Stieltjes integrable functions?

Some examples of Non Stieltjes integrable functions include the Dirichlet function, which is equal to 1 for rational numbers and 0 for irrational numbers, and the Heaviside step function, which is equal to 0 for negative numbers and 1 for positive numbers. These functions are not continuous and do not have a well-defined Stieltjes integral.

How do Non Stieltjes integrable functions differ from Stieltjes integrable functions?

Non Stieltjes integrable functions differ from Stieltjes integrable functions in several ways. First, they may not be continuous or bounded, whereas Stieltjes integrable functions must be both of these. Additionally, Non Stieltjes integrable functions may not have a well-defined Stieltjes integral, whereas Stieltjes integrable functions do.

What are the applications of Non Stieltjes integrable functions?

Non Stieltjes integrable functions have applications in various areas of mathematics, such as in the theory of probability and stochastic processes. They are also used in the study of non-differentiable functions and in the construction of counterexamples in analysis. Additionally, they have applications in physics and engineering, particularly in signal processing and control theory.

How are Non Stieltjes integrable functions useful in mathematics?

Non Stieltjes integrable functions are useful in mathematics because they provide a counterexample to the idea that all functions can be integrated using the Stieltjes integral. They also help to expand our understanding of different types of integrals and their properties. In addition, Non Stieltjes integrable functions are used to construct various mathematical models and are essential in the study of more complex mathematical concepts.

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