Questions tagged [gauge-integral]
For questions about Henstock-Kurzweil integral or gauge integral.
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Is there a Hilbert space of Henstock–Kurzweil square-integrable integrable functions?
As is well-known, the space of square-integrable functions (say, on $[0,\,1]$) where the integral is a Riemann integral is not complete. If one completes it, one obtains the $L^{2}([0,\,1])$ Hilbert ...
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Gauge integral on infinite-dimensional Banach space and differentiability
Call $f:I\to F$ gauge integrable where $I = [a, b]$ is a compact interval and $F$ is a Banach space, if the usual definition holds like if $F = \mathbb{R}$, just replace absolute value by norm. How ...
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Integration by substitution, monotone version for Henstock-Kurzweil integrals
Let $\mathcal{HK}(I)$ denote the Henstock-Kurzweil integrable functions on $I$. By mimicking the case for Lebesgue integral I've proven the following:
Theorem $1$. Let $F$ be an indefinite integral of ...
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Is the set of singularities of a gauge integral always a null set?
Let $\mathcal{R}^*(I)$ be the set of gauge integrable (generalized Riemann integrable) functions $f:I\to \mathbb{R}$ where $I = [a, b]$.
Say that a point $c\in [a, b]$ is a singular for $f\in \mathcal{...
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Bartle's Modern theory of integration, theorem 9.1
Bartle's book, Modern theory of integration, focuses on Henstock-Kurzweil integrals, where $\mathcal{R}^*(I)$ denotes the set of Henstock-Kurzweil integrable functions on $I = [a, b]$.
Also, $\mathcal{...
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Are all Henstock-Kurzweil integrable functions expressible as the sum of a Lebesgue and an improper Riemann integrable function?
This question is based on this post, where in the comments, Toby Bartels conjectures that every Henstock-Kurzweil (gauge) integrable function $f\in\mathcal{HK}$ can be expressed as $f= g + h$ for a ...
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On the set of points where a Henstock-Kurzweil integrable function fails to be Lebesgue integrable
One example of a function that is Henstock-Kurzweil integrable but not Lebesgue integrable is $f(x) = \frac{1}{x} \cos\left(\frac{1}{x^2}\right)$ on $[0, 1]$. However, $f$ only fails to be locally-...
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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 ...
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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, ...
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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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Is every "almost everywhere derivative" Henstock–Kurzweil integrable?
It is well known that the Henstock–Kurzweil integral fixes a lot of issues with trying to integrate derivatives. The second fundamental theorem of calculus for this integral states:
Given that $f : [...
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Does the law of large numbers hold with the Henstock–Kurzweil integral?
If I am understanding, the law of large numbers correctly, one implication is this:
Let $\lambda$ be the Lebesgue measure on $[0,1]$. Let $f_1,f_2,\dots,$ be functions, $[0,1] \rightarrow \mathbb{R}$, ...
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Prove that if $|f|$ is gauge integrable then so is $f$.
I have tried to prove this using the Cauchy criterion for HK-integration but I have been unsuccessful thus far. I have various fancy theorems at my disposal such the MCT and the DCT but I can not ...
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