Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
New page: Henstock integral.
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).
Yeah, I think so. Although Hake’s theorem may not be true in quite as strong a way for the Lebesgue integral either.
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
(Well, Wikipedia didn't specify the endpoints, so I did the integral on Wolfram Alpha.)
I added the example that Toby gave, but as an integral from 0 to - 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.
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.)
@Toby
Might it be worth asking
Are there any functions that are Henstock integrable but not locally Lebesgue integrable?
on MathOverflow?
I thought that I'd try to figure it out myself first!
Actually, the answer is already Yes, in that isn't locally Lebesuge integrable at , 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.)
1 to 9 of 9