A reasonable method of defining an integral that includes the HK integral is to say a Schwartz distribution $f$ is integrable if it is the distributional derivative of a continuous function $F$. Then the integral $(D)\int^b_a f=F(b)-F(a)$. The resulting space of integrable distributions is a Banach space that includes the space of HK integrable functions and is isometrically isomorphic (with Alexiewicz norm) to the continuous functions vanishing at $a$ (with uniform norm).
If $F=C$ is the Cantor(the Devil's staircase) function and $\langle C'\rangle$ (we use notation $\langle C'\rangle$ to avoid confusion and in some situation $C'$) is the distributional derivative of $C$, then
$(D)\int^0_1 \langle C'\rangle=C(1)-C(0)=1-0=1$. Note that $\langle C'\rangle$ is a measure.
If here $C'$ denotes derivative in classical sense then $C'=0$ a.e. and $(HK)\int^0_1 C'=0$.
Suppose $F$ is continuous on $[a,b]$. Also suppose $f(t)=F'(t)$ exists except on a countable set $Q=(c_k)$; define $f$ arbitrarily on $Q$. Then
Then $\int_a^t f(x) dx $ exists and equals $F(t)-F(a)$.
See for example
1) See https://math.vanderbilt.edu/schectex/ccc/gauge/letter/
"An Open Letter to Authors of Calculus Books". Retrieved 27 February 2014.
2) https://www.fsb.unizg.hr/matematika/sikic
https://www.fsb.unizg.hr/matematika/download/ZS_NL_and_Henstock.pdf
NEWTON–LEIBNIZ FORMULA AND HENSTOCK–KURZWEIL INTEGRAL ZVONIMIR \v SIKI\'C, ZAGREB
Dear Miodrag,
Fist here is definition of the Henstock–Kurzweil integral :
The set of KH-integrable functions forms an ordered vector space and the integral is a positive linear form on this space. On a segment, any Riemann-integrable function is KH-integrable (and of the same integral).
For relation between the gauge integral and the Lebesgue and Riemann integrals, i suggest you to see links and attached file on topic.
https://arxiv.org/pdf/1609.05454.pdf
https://math.vanderbilt.edu/schectex/ccc/gauge/
https://www.worldscientific.com/worldscibooks/10.1142/7933
Best regards
For $c\in \mathbb{R}$ distribution $I_c$ defined by $I_c(\varphi)=c \int \varphi$, $\varphi\in D (\mathbb{R})$, is called constant distribution.
Instead of $I_c$ we often use the notation $c$ if there is no possibility of confusion.
The following property is crucial for the definition of the (D)-integral:
(A) If $F_1$ and $F_2$ are two primitive distribution of a distribution $f$, then $F_2=F_1 +I_c$. In addition if $F_1$ is continuous then $F_2$ is also continuous and $F_2=F_1 + c$.
A reasonable method of defining an integral that includes the HK integral is to say a Schwartz distribution $f$ is integrable if it is the distributional derivative of a continuous function $F$. Then the (D)-integral is defined by $(D)\int^b_a f=F(b)-F(a)$. The property (A) shows that this definition is correct.
The resulting space of integrable distributions is a Banach space that includes the space of HK integrable functions and is isometrically isomorphic (with Alexiewicz norm) to the continuous functions vanishing at $a$ (with uniform norm).
لإhe Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral , Luzin integral or Perron integral, but not to be confused with the more general wide Denjoy integral – is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral, and in some situations is more general than the Lebesgue integral. In particular, a function is Lebesgue integrable if and only if the function and its absolute value are Henstock–Kurzweil integrable.
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