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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMar 2nd 2014
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   Author: Mike Shulman
   Format: MarkdownItexNew page: [[Henstock integral]].

   New page: Henstock integral.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeMar 2nd 2014
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   Author: Todd_Trimble
   Format: MarkdownItexIs it the first theorem that explains why the Henstock integral behaves "better" than the Lebesgue integral? From memory, it's absolutely continuous functions which have Lebesgue integrable derivatives (almost everywhere).

   Is it the first theorem that explains why the Henstock integral behaves “better” than the Lebesgue integral? From memory, it’s absolutely continuous functions which have Lebesgue integrable derivatives (almost everywhere).

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMar 2nd 2014
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   Author: Mike Shulman
   Format: MarkdownItexYeah, I think so. Although Hake's theorem may not be true in quite as strong a way for the Lebesgue integral either.

   Yeah, I think so. Although Hake’s theorem may not be true in quite as strong a way for the Lebesgue integral either.

4. • CommentRowNumber4.
   • CommentAuthorTobyBartels
   • CommentTimeMar 5th 2014
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   Author: TobyBartels
   Format: MarkdownItexRight, a function can be improperly Lebesgue integrable (and therefore Henstock integrable) without being Lebesgue integrable. (Such a function must alternate sign a lot.) Wikipedia gives the example of $$ \int_{-1}^1 \frac{\sin (1/x^3)}{x} = \frac{1}{3} \pi - \frac{2}{3} Si(1) \approx 0.416476 .$$ (Well, Wikipedia didn\'t specify the endpoints, so I did the integral on Wolfram Alpha.)

   Right, a function can be improperly Lebesgue integrable (and therefore Henstock integrable) without being Lebesgue integrable. (Such a function must alternate sign a lot.) Wikipedia gives the example of

   $$\int_{-1}^1 \frac{\sin (1/x^3)}{x} = \frac{1}{3} \pi - \frac{2}{3} Si(1) \approx 0.416476 .$$

   (Well, Wikipedia didn't specify the endpoints, so I did the integral on Wolfram Alpha.)

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeFeb 20th 2015
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   Author: DavidRoberts
   Format: MarkdownItexI added the example that Toby gave, but as an integral from 0 to $x$ - as the integrand is even, one can easily figure out what happens for intervals with negative endpoint(s). I too am trusting WolframAlpha, but I would prefer a reference (or perhaps we can do it as an exercise!) that does it by hand.

   I added the example that Toby gave, but as an integral from 0 to $x$ - as the integrand is even, one can easily figure out what happens for intervals with negative endpoint(s). I too am trusting WolframAlpha, but I would prefer a reference (or perhaps we can do it as an exercise!) that does it by hand.

6. • CommentRowNumber6.
   • CommentAuthorTobyBartels
   • CommentTimeMay 18th 2023
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   Author: TobyBartels
   Format: MarkdownItexExamples, along with some subheaders and standardization of notation. <a href="https://ncatlab.org/nlab/revision/diff/Henstock+integral/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/Henstock+integral/8">v8</a>, <a href="https://ncatlab.org/nlab/show/Henstock+integral">current</a>

   Examples, along with some subheaders and standardization of notation.

   diff, v8, current

7. • CommentRowNumber7.
   • CommentAuthorTobyBartels
   • CommentTimeMay 18th 2023
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   Author: TobyBartels
   Format: MarkdownItexOh, and regarding checking the example that I did with WolframAlpha: once you know the semidefinite integral (so thanks, David, for getting that), it\'s easy to check by differentiating (and relying on the FTC). (So I did that but didn't put it in the article since at that point it\'s kind of trivial.)

   Oh, and regarding checking the example that I did with WolframAlpha: once you know the semidefinite integral (so thanks, David, for getting that), it's easy to check by differentiating (and relying on the FTC). (So I did that but didn’t put it in the article since at that point it's kind of trivial.)

8. • CommentRowNumber8.
   • CommentAuthorDavidRoberts
   • CommentTimeMay 19th 2023
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   Author: DavidRoberts
   Format: MarkdownItex@Toby Might it be worth asking >Are there any functions that are Henstock integrable but not locally Lebesgue integrable? on MathOverflow?

   @Toby

   Might it be worth asking

   Are there any functions that are Henstock integrable but not locally Lebesgue integrable?

   on MathOverflow?

9. • CommentRowNumber9.
   • CommentAuthorTobyBartels
   • CommentTimeMay 19th 2023
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   Author: TobyBartels
   Format: MarkdownItexI thought that I\'d try to figure it out myself first! Actually, the answer is already Yes, in that $x \mapsto \sin(1/x^3)/x$ isn\'t locally Lebesuge integrable at $0$, but it\'s locally Lebesgue integrable everywhere else, which is enough to define all of its integrals as improper Lebesgue integrals. So I think that the relevant question is whether they\'re locally Lebesgue integrable with an isolated set of exceptions. (An even better example would be one that\'s Lebesgue integrable nowhere, but that\'s even less plausible.)

   I thought that I'd try to figure it out myself first!

   Actually, the answer is already Yes, in that $x \mapsto \sin(1/x^3)/x$ isn't locally Lebesuge integrable at $0$, but it's locally Lebesgue integrable everywhere else, which is enough to define all of its integrals as improper Lebesgue integrals. So I think that the relevant question is whether they're locally Lebesgue integrable with an isolated set of exceptions. (An even better example would be one that's Lebesgue integrable nowhere, but that's even less plausible.)