Lebesgue Integration: Right-Continuous Function & Series Convergence

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Discussion Overview

The discussion revolves around the Lebesgue integration concerning right-continuous functions and the implications of series convergence, particularly focusing on the necessity of absolute convergence in the context of jump components.

Discussion Character

  • Technical explanation, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant inquires about the necessity of absolute convergence for series when considering Lebesgue integration with respect to right-continuous functions, specifically regarding jump components.
  • Another participant expresses confusion about the term "series part" and requests clarification or an example related to the inquiry.
  • A third participant suggests that the discussion may pertain to the Stieltjes-Lebesgue measure.
  • A fourth participant confirms that the topic relates to the Lebesgue-Stieltjes integral, indicating that the function involved is right-continuous.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the necessity of absolute convergence or the specifics of the series part, and there is some confusion regarding the terminology used.

Contextual Notes

The discussion includes unresolved questions about the definitions and implications of absolute versus simple convergence in the context of Lebesgue integration.

wayneckm
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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:
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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.
 
I think... he's talking about the Stieltjes-Lebesgue measure df(x)?
 
Yup, it should be Lebesgue-Stieltjes integral [tex]\int f(x) d g(x)[/tex] with [tex]g(x)[/tex] being right-continuous.
 

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