Integral more general then Lebesgue integral?

In summary, the Lebesgue integral is a specific type of integral that is defined for measurable functions. However, there are more general integrals that can be defined for a larger class of functions. If these integrals do not have the same properties as the Lebesgue integral, they may not be as useful. Some examples of more general integrals include the gauge integral and the Henstock-Kurzweil integral, but these may not be as simple or easy to work with as the Lebesgue integral. Measure theory is also a key factor in why the generalized Riemann integral is not commonly used.
  • #1
r4nd0m
96
1
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
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
I seem to recall reading something in Pugh's Real Mathematical Analysis where he described some integration theories more general than Lebesgue's.
 
  • #4
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
And what if we changed open sets in the definition of a measurable function to some more general sets? What would be wrong?
 
  • #6
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
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
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).
 

1. What is the difference between Integral and Lebesgue integral?

The Integral is a mathematical concept that represents the area under a curve, while the Lebesgue integral is a more general concept that extends the Integral to a wider class of functions.

2. What are the advantages of using Lebesgue integral over the Integral?

One of the main advantages of using Lebesgue integral is its ability to handle a wider range of functions, including non-continuous and unbounded functions. It also provides a more abstract and rigorous approach to integration.

3. How is the Lebesgue integral defined?

The Lebesgue integral is defined using the concept of measure theory, which is a mathematical theory that deals with the concept of size or volume in a more abstract setting. It involves partitioning the domain of the function into smaller intervals and calculating the "size" of these intervals using a measure function.

4. What are some practical applications of Lebesgue integral?

Lebesgue integral has many applications in mathematics, physics, and engineering. It is used in probability theory, Fourier analysis, and signal processing. It also has applications in economics, finance, and statistics.

5. Are there any limitations to using Lebesgue integral?

One limitation of Lebesgue integral is that it requires a more advanced mathematical background, including measure theory, to fully understand and use it. It can also be more computationally intensive compared to the regular Integral.

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