Lebesgue Integration: Right-Continuous Function & Series Convergence

[wayneckm]

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.

 

[mathman]

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.

 

[Hurkyl]

I think... he's talking about the Stieltjes-Lebesgue measure df(x)?

 

[wayneckm]

Yup, it should be Lebesgue-Stieltjes integral $$\int f(x) d g(x)$$ with $$g(x)$$ being right-continuous.