Riemann integrability of f composites g, given f integrable,

[passionate]

I'm having trouble with this problem, and it's one of the review problems for my upcoming test. Hope someone can help.

Let g: [c,d] into [a,b] be continuous and for every zero set Z in [a,b], pre-image g of Z is a zero set in [c,d].If f is Riemann integrable, show that g(f) is Riemann integrable.

 

[pka]

What is the definition of a 'zero set'?

 

[passionate]

What is the definition of a 'zero set'?

A set Z tapcon.gif R is a zero set if and only if for each [epsilon] > 0, there is a countable covering of Z by open interval (a,b) such that [sum] b_i - a_i [/sum] less than or euqal to [epsilon] , i goes from 1 to infinity.

 

[pka]

There are a huge differences the in approaches to the Riemann Integral. The use of the ‘zero set’ seems to indicate that you have not done measure theory. So it is difficult to know if what I have to say will help you at all.

The fact that f is Riemann integrable means that any set of discontinuities of f must be a zero set. You are given that g is continuous and g-inverse images of zero sets are zero sets. Therefore, the composition gof must be continuous almost-everywhere. That is enough to tell you that gof is Riemann integrable.

Again your text/instructor may well expect some talk about ‘bounded variation’ or limits of step-functions.

I hope that is of some help to you.

 

[passionate]

There are a huge differences the in approaches to the Riemann Integral. The use of the ‘zero set’ seems to indicate that you have not done measure theory. So it is difficult to know if what I have to say will help you at all.

The fact that f is Riemann integrable means that any set of discontinuities of f must be a zero set. You are given that g is continuous and g-inverse images of zero sets are zero sets. Therefore, the composition gof must be continuous almost-everywhere. That is enough to tell you that gof is Riemann integrable.

Again your text/instructor may well expect some talk about ‘bounded variation’ or limits of step-functions.

I hope that is of some help to you.

Thanks a lot for your help. I am supposed to use the Riemann-Lebesgue Theorem on this problem, which I totally forgot. Finally, I am able to show that i]gof[/i] is bounded, and its discontinuity set is a zero set. I should have seen it right away if I wasn't that stupid. Anyway, I really appreciate your help.

 

[pka]

Thank you for your response.
I can only hope that this is a lesson to all who post in “Advanced Topics”.
Had you mentioned “Riemann-Lebesgue Theorem”, I may have had some clearer idea of the ‘machinery’ with which you can work.
Please tell us where you are! Don’t let us guess!