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# Questions tagged [gauge-integral]

For questions about Henstock-Kurzweil integral or gauge integral.

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### Henstock-Kurzweil integral and the Controlled Convergence Theorem

Let me recall that, for the Henstock-Kurzweil integral, there holds a very general convergence theorem: Let $(f_n)$ be a sequence of $HK$-integrable functions on $[0, 1]$ such that $f_n(x) \to f(x)$ ...
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### What makes Cousin's theorem remarkable?

I've stumbled upon a mention of Cousin's theorem in the context of Henstock–Kurzweil integral and got confused. I do not understand why this fact is called a theorem and what makes it any remarkable, ...
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### Some questions concerning the construction of the line integral in $\mathbb{C}$

I am comparing two ways of defining the integral along a path of a function of complex domain and value. One is the one given in Conway's Functions of one Complex Variable and the other is given in ...
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### If $f(x)$ is Henstock-Kurzweil integrable on $[a,b]$, then is $f(x)\mathrm{e}^{\mathrm{i}x}$ also Henstock-Kurzweil integrable on $[a,b]$?

I was wondering about how Fourier series behaves in the setting of Henstock-Kurzweil integration. For example, the non-Lebesgue-integrable function $f(x) = \dfrac{1}{x}\mathrm{e}^{\mathrm{i}/x}$ can ...
159 views

### Is there a constructive presentation of the Henstock-Kurzweil integral?

Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
48 views

### Can the LCT and MCT for Lebesgue integrable functions be viewed as a lattice completeness result?

The set of Lebesgue integrable functions form a lattice under pointwise min and max (also more generally for R, Henstock-Kurzweil integrable functions with an upper or lower bound form a lattice as ...
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### Mistake in Bartle's proof of Hake's Theorem?

Here is Bartle's proof of Hake's Theorem found in "A Modern Theory of Integration". I think there is a mistake in the highlighted line: The Theorem: $f:[a,b]\to \mathbb{R}$ is gauge integrable if and ...
185 views

### Proof that if $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$

I am asking for a self contained proof of this assertion: If $f,g,h:[a,b]\to \mathbb{R}$ with $h\le f,g$ and $f,g,h$ are gauge integrable then so is $\min(f,g)$. The integral in question is the ...
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### Integrability: Neither improper Riemann nor Lebesgue but Henstock-Kurzweil

Can you think of a function that is neither improper Riemann nor Lebesgue integrable, but is Henstock-Kurzweil integrable? I'd like to put a bounty on this question, but my reputation is not nearly ...