Fubini's Theorem: Integral Existence for Non-Continuous Functions

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SUMMARY

Fubini's Theorem addresses the interchange of integration order in double integrals, specifically when the integrand is continuous. However, the existence of an integral is not solely dependent on the continuity of the function; a function can still be integrable if the set of its discontinuities has measure zero. Therefore, the assertion that a non-continuous function cannot have an integral is incorrect. Understanding these nuances is crucial for proper application of Fubini's Theorem in analysis.

PREREQUISITES
  • Understanding of Fubini's Theorem in analysis
  • Knowledge of measure theory, specifically measure-zero sets
  • Familiarity with concepts of integrability in mathematical analysis
  • Basic proficiency in handling double integrals
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  • Study the implications of measure-zero sets on integrability
  • Explore advanced applications of Fubini's Theorem in multiple integrals
  • Learn about Lebesgue integration and its relationship to continuity
  • Investigate counterexamples of non-continuous functions that are integrable
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Mathematics students, educators, and researchers focusing on real analysis, particularly those studying integration techniques and the properties of functions in relation to Fubini's Theorem.

Pearce_09
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hello,
I posted a question a while ago about fubini's theorem, and i believe i have found my answer.. but to clear things up, i have one more question.

If a function f(x) is not continuous then the integral (by fubinis) does not exist.
is this correct. ( I believe it is correct, but i would feel better if someone else agreed as well)
thank you!
adam
 
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IIRC, a (dumbed down) version of Fubini in analysis says that the order of integration in a double integral may be interchanged if the integrand is a continuous function. That doesn't mean the integral doesn't exist when if the integrand is not continuous.

The LateX images in your other post are not showing, so we can't help you there.
 
A function is integrable iff the set of discontinuities is a measure-0 set. Fubini's theorem has little to do with this - it's about double integration.
 

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