Integral more general then Lebesgue integral?

  • Thread starter r4nd0m
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  • #1
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integral more general than the Lebesgue integral?

The Lebesgue integral is defined for measurable functions. But isn't it possible to define a more general integral defined for a larger class of functions?
I guess that we would then loose some of the fine properties of the Lebesgue integral - but which and why?
 
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  • #2
HallsofIvy
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I'm not sure what you mean. There is "The" Lesbeque integral and a wide class of "Lebesque" integrals. The latter involves all methods of putting a "measure" on sets that preserve "sigma-additivity" and "translation invariance". Those are what you would lose if you used any more "general" definition of measure- and they are fairly important!
 
  • #3
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I seem to recall reading something in Pugh's Real Mathematical Analysis where he described some integration theories more general than Lebesgue's.
 
  • #4
Gib Z
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But when the functions they describe lose the basic required properties of the Lebesgue integral, the integrals become harder to define and less useful.
 
  • #5
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And what if we changed open sets in the definition of a measurable function to some more general sets? What would be wrong?
 
  • #6
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Just stick to the definition of integrability in the sense of wide class Lebesque
integrability and you are ok.Otherwise you'll run into problems and possibly paradoxes!
 
  • #7
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The gauge integral (and is variations) includes the Lebesgue integral as a special case. It is equivalent for bounded functions on a finite interval. It can also integrate some unbounded functions and some functions that are not absolutely integrable. Its defintion is nearly as simple as the Riemann integral.

http://en.wikipedia.org/wiki/Henstock–Kurzweil_integral
 
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  • #8
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As others have mentioned over three years ago, measure theory is one reason why the generalized Riemann integral is not used. Also, part of the utility of the Lebesgue theory seems to lie in the theorems that allow the interchange of limits and integrals, namely Fatou's Lemma, Monotone Convergence Theorem, and the Dominated Convergence theorem (which by the way are simple and nice results once measure theory is developed).
 

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