Integral over a set of measure 0

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Discussion Overview

The discussion centers around the properties of integrals over sets of measure zero, particularly in the context of Lebesgue and Riemann integration. Participants explore whether the integral of a function defined on a set of measure zero is always zero, and under what conditions it may be undefined or lead to inconsistencies.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants question if the integral over a set of measure zero is always zero and whether it can be undefined.
  • Others argue that for ordinary functions, the integral will be zero, but it may be undefined for functions that take on infinite values.
  • A participant mentions that in Lebesgue integration, the convention \(\infty \cdot 0 = 0\) is typically adopted, suggesting that even if a function takes on infinite values on a measure zero set, the integral would still be zero.
  • Another participant asserts that the integral over a measure zero set is a trivial theorem in Lebesgue integration, as all simple functions have an integral of zero.
  • One participant raises a concern about potential inconsistencies if \(\infty \cdot 0\) is not defined as zero, using an example involving approximating a function with a sequence of functions defined on a measure zero set.
  • Further discussion involves the implications of defining functions over sets of non-zero measure that converge to zero, questioning the nature of the functions being integrated.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the integral over a set of measure zero is always zero or if it can be undefined. Multiple competing views remain regarding the treatment of infinite values in this context.

Contextual Notes

There are unresolved assumptions regarding the definitions of functions and the implications of integrating over sets of varying measures. The discussion reflects differing interpretations of integration conventions and their mathematical consequences.

Boot20
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Is the integral over a set of measure zero always equals to zero? Can the integral be undefined?
 
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It may be undefined if the function itself is peculiar with infinity as its value. For ordinary functions the integral will be 0.
 
Last edited:
mathman said:
It may be undefined if the function itself is peculiar with infinity as its value. For ordinary functions the integral will be 0.

Sorry, I have the question that, if Lebesgue integration, they always define the convention \infty \cdot 0 = 0, so, in this case, even the function takes \infty in a set of measure 0, the integral is still 0?
 
Convention is an easy way out.
 
For Lebesgue integration, that the integral over a set of measure zero is a rather trivial theorem, following from the fact that all simple functions have integral zero -- so via (what I believe is) the usual formulation, it doesn't even need to be treated as a special case.


Riemann integration assumes the function is real-valued, so it doesn't even apply if you are considering extended-real-number-valued functions that take on the values +\infty or -\infty.
 
Last edited:
But apparently to me \infty \cdot 0 = 0 should be adopted
Else, if f admit \infty on set A of measure 0, we may use f_{n} = n on A to approximate f from below, then, the integral of f_{n} is zero, by monotone convergence theorem, the integral of f should be zero as well. If we do not define \infty \cdot 0 = 0, we may get inconsistency in this case?
 
wayneckm said:
But apparently to me \infty \cdot 0 = 0 should be adopted
Else, if f admit \infty on set A of measure 0, we may use f_{n} = n on A to approximate f from below, then, the integral of f_{n} is zero, by monotone convergence theorem, the integral of f should be zero as well. If we do not define \infty \cdot 0 = 0, we may get inconsistency in this case?

But what if the f_{n} are defined over sets of measure non-zero, but that the sum of the measure of those sets converges to zero?
 
Last edited:
Boot20 said:
But what if the f_{n} are defined over sets of measure noe intgen-zero, but that the sum of the measure of those sets converges to zero?

Sorry that I cannot get your idea exactly. Or can you tell me explicitly what is the function f to be integrated?
 

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