Can the Lebesgue Integral of a_p be Defined for Non-Differentiable Functions?

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Discussion Overview

The discussion revolves around the Lebesgue integral of a specific function defined on integers, particularly focusing on its behavior for non-differentiable functions. Participants explore the implications of defining integrals for functions that are non-zero only at discrete points and compare it with other functions, raising questions about the nature of Lebesgue integration.

Discussion Character

  • Debate/contested
  • Mathematical reasoning
  • Conceptual clarification

Main Points Raised

  • One participant defines the function a_{p}(x) as 1 for prime integers and 0 elsewhere, questioning its Lebesgue integral over a finite interval.
  • Another participant argues that since the function is non-zero only at a finite number of points, the integral must be zero.
  • Some participants assert that the Riemann and Lebesgue integrals of the function are trivially zero on any interval, questioning the need for the initial inquiry.
  • A participant raises a comparison with the function that is 1 for rational numbers and 0 elsewhere, questioning why its integral is not zero.
  • Responses clarify that functions defined on sets of measure zero yield an integral of zero, which is a key aspect of Lebesgue theory.
  • There is a suggestion that if the function is defined as 1 for irrational numbers and 0 for rational numbers, the integral over an interval would equal the length of that interval.
  • Another participant brings up doubts regarding Lebesgue integration, asking about the relationship between Lebesgue and Riemann integrals for the exponential function and the formula for integration by parts.
  • Concerns are raised about the applicability of integration by parts in Lebesgue integration, suggesting it requires differentiable functions, thus questioning its relevance.

Areas of Agreement / Disagreement

Participants express disagreement regarding the integral values of different functions, particularly the distinction between functions defined on rational versus irrational numbers. The discussion remains unresolved as participants hold varying views on the implications of Lebesgue integration.

Contextual Notes

Participants highlight the importance of measure theory in determining the value of integrals for functions defined on sets of measure zero, but there are unresolved questions about the conditions under which integration by parts applies in the context of Lebesgue integration.

eljose
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Let define the function:

[tex]a_{p}(x)= 1[/tex] if x is an integer and prime and 0 elsewhere, my

question is...what would be its Lebesgue integral let,s say from [c,d] with c and d positive and real..
 
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Since this function is only non-zero on only a finite number of points between c and d, isn't the integral obviously 0?
 
It's Riemann (and hence lebesgue) integral is rather trivially zero on any interval. My question is why would you need to ask this?
 
then why the integral of the function [tex]f(x)=1[/tex] iff x is rational and 0 elsewhere is different from 0?...
 
eljose said:
then why the integral of the function [tex]f(x)=1[/tex] iff x is rational and 0 elsewhere is different from 0?...

You're sayig the Lebesgue integral of this function is non zero? How do you figure?
 
1 on a set of measure zero 0 every where else, aka almost everywhere zero. that the integral is zero of such a thing is practically the point of lebesgue theory.
 
eljose said:
then why the integral of the function [tex]f(x)=1[/tex] iff x is rational and 0 elsewhere is different from 0?...

It isn't! Who told you that it was? The lebesque integral of the function you give is 0 over any finite interval.

IF, instead, you define f(x)= 1 if x is irrational and 0 if x is rational (1- your f(x)) then the integral of f over the interval [a, b] is b-a.
 
i know i have posted this topic or analogue before but i have the doubts with lebesgue integration:

a) the Lebesgue integral of exp(x)..is equal to Riemann integral of exp(x)

b) [tex]D_{t}\int_{0}^{t}d\mu{f}= f ?[/tex]

c)what would be the formula for integration by parts in Lebesgue integration?..

thanks.
 
integration by parts requires the integrands to be differentiable or to be a derivative and hence continuous, so there is no point in using lebesgue integration, is there?
 

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