Lebesgue Criterion for Riemann Integrability

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In summary, the Lebesgue Criterion for Riemann Integrability is a theorem that states a function is Riemann integrable if and only if its set of discontinuities has measure zero. It differs from the Riemann Criterion by only requiring the set of discontinuities to have measure zero, not the function itself to be bounded. The criterion can be used to prove a function is not Riemann integrable, but it does not provide a method for determining if a function is. The advantages of using the Lebesgue Criterion include its generality and flexibility, but it is limited to functions on closed and bounded intervals and can be difficult to apply in practice.
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quasar987
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There's a theorem in my real analysis textbook that says

A function f is Riemann-integrable iff the set of its points of discontinuity is of measure zero.But take say f(x)=1/x. It is only discontinuous as x=0, but it's not integrable on (-e,e). :grumpy:
 
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  • #2
i could be wrong but i think that theorem only applies to bounded functions.
 
  • #3
This looks like a blatant omission of the word "bounded".
 

1. What is the Lebesgue Criterion for Riemann Integrability?

The Lebesgue Criterion for Riemann Integrability is a mathematical theorem that provides a necessary and sufficient condition for a function to be Riemann integrable. It states that a function is Riemann integrable if and only if its set of discontinuities has measure zero.

2. How is the Lebesgue Criterion different from the Riemann Criterion for Integrability?

The Riemann Criterion for Integrability states that a function is Riemann integrable if and only if it is bounded and its set of discontinuities has measure zero. The Lebesgue Criterion is more general because it only requires the set of discontinuities to have measure zero, not the function itself to be bounded.

3. Can the Lebesgue Criterion be used to determine if a function is not Riemann integrable?

Yes, the Lebesgue Criterion can be used to prove that a function is not Riemann integrable. If a function does not satisfy the criterion, then it is not Riemann integrable. However, the criterion does not provide a method for determining if a function is Riemann integrable, only if it is not.

4. What are the advantages of using the Lebesgue Criterion for Riemann Integrability?

The Lebesgue Criterion offers a more general and powerful tool for determining Riemann integrability compared to the Riemann Criterion. It also allows for more flexibility in choosing integration intervals and can handle a wider range of functions, including unbounded ones.

5. Are there any limitations to the Lebesgue Criterion for Riemann Integrability?

One limitation of the Lebesgue Criterion is that it only applies to functions defined on a closed and bounded interval. It cannot be used to determine the Riemann integrability of functions defined on unbounded intervals. Additionally, the criterion can be difficult to apply in practice due to the concept of measure and the need for advanced mathematical tools.

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