Lebesgue Integration: Right-Continuous Function & Series Convergence

In summary, there was a discussion about the Lebesgue integration with respect to a right-continuous function and the inclusion of a series part to account for jump components. The question of whether the series must be absolutely convergent was raised, and the reason for defining absolute convergence instead of simply convergence was questioned. The conversation also touched on the Stieltjes-Lebesgue measure and the Lebesgue-Stieltjes integral with a right-continuous function.
  • #1
wayneckm
68
0
Hello all,

I would like to know when the Lebesgue integration w.r.t. a right-continuous function, there would be a series part which takes account of the jump components.

Is it true that we require the series to be absolutely convergent? if so, what is the rationale of defining this instead of simply convergent?

Thanks very much.
 
Last edited:
Physics news on Phys.org
  • #2
wayneckm said:
Hello all,

I would like to know when the Lebesgue integration w.r.t. a right-continuous function, there would be a series part which takes account of the jump components.

Is it true that we require the series to be absolutely convergent? if so, what is the rationale of defining this instead of simply convergent?

Thanks very much.

The question is confusing (series part?). Could you give an example of what you are asking about.
 
  • #3
I think... he's talking about the Stieltjes-Lebesgue measure df(x)?
 
  • #4
Yup, it should be Lebesgue-Stieltjes integral [tex] \int f(x) d g(x) [/tex] with [tex] g(x) [/tex] being right-continuous.
 

Related to Lebesgue Integration: Right-Continuous Function & Series Convergence

1. What is the Lebesgue integral?

The Lebesgue integral is a mathematical concept that extends the idea of integration from traditional calculus. It allows for the integration of a wider range of functions, including those that are not continuous or have discontinuities. It also provides a more general and rigorous approach to integration.

2. What is a right-continuous function?

A right-continuous function is a type of function that has a limit from the right at every point on its domain. This means that the value of the function at a particular point is equal to the limit of the function as the input approaches from the right side of that point.

3. What is the significance of right-continuous functions in Lebesgue integration?

Right-continuous functions are important in Lebesgue integration because they allow for the integration of functions that have jump discontinuities. This means that the Lebesgue integral can be used to integrate a wider range of functions compared to traditional integration methods.

4. What is series convergence in Lebesgue integration?

Series convergence in Lebesgue integration refers to the convergence of a series of functions that are being integrated. This is important because it allows for the integration of functions that cannot be integrated individually, but can be integrated when combined in a series.

5. How does Lebesgue integration differ from Riemann integration?

Lebesgue integration differs from Riemann integration in the types of functions that can be integrated. Riemann integration is limited to continuous functions, while Lebesgue integration allows for the integration of a wider range of functions, including those with jump discontinuities. Additionally, Lebesgue integration provides a more general and rigorous approach to integration compared to Riemann integration.

Similar threads

I On convolution theorem of Laplace transform: Schiff
    • Calculus
Replies
4
Views
918
Insights A Novel Technique of Calculating Unit Hypercube Integrals
    • Calculus
Replies
1
Views
2K
I Testing for Divergence using the Integral Test
    • Calculus
Replies
1
Views
1K
MHB Can the Series Sum Be Expressed as an Integral as N Approaches Infinity?
    • Calculus
Replies
3
Views
1K
I Did Math DF's integral calculator make a glaring mistake?
    • Calculus
Replies
8
Views
395
A Memory issues in numerical integration of oscillatory function
    • General Math
Replies
3
Views
614
I What is convergence and 1+2+3+4....... = -1/12
    • Calculus
2
Replies
37
Views
3K
Show function series involving arctan is not differentiable at x=0
    • Calculus and Beyond Homework Help
Replies
26
Views
980
Implications of Lebesgue Integration for Bounded Functions
    • Calculus
Replies
3
Views
3K
MHB Series Convergence and Divergence II
    • Calculus
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top