On AP–Henstock–Kurzweil Integrals and Non-Atomic Radon Measure

Abstract

The AP–Henstock–Kurzweil-type integral is defined on $\mathtt{X},$ where $\mathtt{X}$ is a complete measure metric space. We present some properties of the integral, continuing the study’s use of a Radon measure $\mu .$ Finally, using locally finite measures, we extend the AP–Henstock–Kurzweil integral theory to second countable Hausdorff spaces that are locally compact. A Saks–Henstock-type Lemma is proved here.

1. Introduction

J. Kurzweil first proposed a solution to the primitives problem in 1957, and R. Henstock did the same in 1963. It is commonly known that the Henstock–Kurzweil integral (HK-integral) is a generalized form of the Riemann integral. J. Kurzweil and R. Henstock each separately produced this generalization. The Henstock–Kurzweil integral has a construction that is comparable to that of the Riemann integral but stronger than the Lebesgue integral. Additionally, it is well known that the HK-integral can resolve the issue of primitives in the real line. The Riemann sums limit over the appropriate integration domain partitions is referred to as the HK-valued integrals. The HK-integral seems to have a constructive definition. Within the HK-integral, a gauge-like positive function is employed to assess a partition’s fineness rather than a constant as in the Riemann integral, which is the fundamental distinction between the two definitions. Cao in [1] introduced the Banach valued Henstock–Kurzweil integral, Boccuto et al. in [2] defined the Henstock–Kurzweil-type integral for functions defined on a (possibly unbounded) subinterval on the extended real line and with values in Banach spaces. A Fubini-type theorem was proved, for the Kurzweil–Henstock integral of Riesz-space-valued functions defined on (not necessarily bounded) subrectangles of the extended real plane (see [3]). The integral of the function close to singular points is better approximated as a result. The problem turns out to be more challenging for the integration of approximative derivatives. Finding relationships between the Denjoy–Khintchine integral and its roughly continuous generalizations and approximate Perron-type integrals was the focus of most studies in this area (see [4] for Denjoy integral). John Burkill [5] was credited with the invention of the roughly continuous Perron integral (AP-integral). According to the uniformly AP integrable in short UAP and element-wise boundedness conditions, Jae Myung Park et al. [6] investigated convergence theorems for the AP-integral. By demonstrating that the AP–Denjoy integral and the AP–Henstock–Kurzweil integral are equal and identical, Jae Myung Park et al. [7] defined the AP-Denjoy integral. In [8], Skvortsov et al. draw attention to results that are powerful than those shown in Jae’s work. They demonstrate how some of them are amenable to formulation in perspective of a derivation basis specified by a local system, of which it is known that the approximate basis is a specific case. They also consider how the $\sigma$-finiteness of the variational measure generated by a function interacts with the conventional idea of generalized bounded variation. The Riemann-type integral is equal to the properly defined Perron-type integral on a large class of bases (see [9]). Skvortsov et al. [10] say only that Burkill’s AP-integral is covered by AP–Henstock–Kurzweil integral. Shin and Yoon [11] introduced the concept of approximately negligible variation and give a necessary and sufficient condition that a function $\mathcal{F}$ be an indefinite integral of an AP–Henstock–Kurzweil integrable function $\mathtt{f}$ on $[a,b].$ The concept of bounded variation is then used to describe the characterization of AP–Henstock–Kurzweil integrable functions. Bongiorno et al. mention in their research [12] a type of Henstock–Kurzweil integral defined on a complete metric space $\mathtt{X},$ using a Radon measure $\mu$ and a family of sets $\mathtt{F}$ which fulfill the covering theorem of Vitali for $\mu .$ In particular, the traditional Henstock–Kurzweil integral on $\mathtt{R},$ where $\mathtt{R}$ is the set of reals, is enclosed by this integral; for more information, see [13]. The construction of the $\mu$–Henstock–Kurzweil integral motivated us to construct ${\mu }_{AP}$–Henstock–Kurzweil integrals on complete metric spaces with a non-atomic Radon measure. In this paper, we analyze the AP-integral of Henstock–Kurzweil type, described on $\mathtt{X},$ possessed by $\mu$ and a class of “sets” $\mathtt{F}$ fulfilling the Vitali’s covering theorem. We finally enlarge this notion in the setting of locally compact second countable Hausdorff spaces with bounded or locally finite measures.

The paper is organized as follows: in Section 2, the basic concepts and terminology are introduced together with some definitions and results. In Section 3, we introduce the $AP$–Henstock–Kurzweil-type integral called ${\mu }_{AP}$–Henstock–Kurzweil integral of a set-valued function with respect to a Radon measure. Simple properties of ${\mu }_{AP}$–Henstock–Kurzweil integrals are discussed in Section 3.1. The relationships between ${\mu }_{AP}$–Henstock–Kurzweil integrable functions and Lebesgue integrable functions are discussed in Proposition 4. In Section 4, we extend the theory of the ${\mu }_{AP}$–Henstock–Kurzweil integral to locally finite measures on locally compact second countable Hausdorff spaces. A few fundamental results are discussed in this section. The main result in this section is the Saks–Henstock-type Lemma on Theorem 12.

2. Preliminaries

Let us fix $\mathtt{X}=(\mathtt{X},d)$ as a Cauchy metric space with a non-atomic Radon measure $\mu .$ Throughout the paper, the complete metric spaces or the Cauchy metric spaces will be termed as Cauchy spaces. A $\sigma$-algebra is a collection $\mathtt{M}$ of subsets of $\mathtt{X}$ satisfying the conditions:

• $\mathtt{X}$ is in $\mathtt{M}.$
• $\mathtt{A}$ is in $\mathtt{M}$ implies $\mathtt{X}\backslash \mathtt{A}\in \mathtt{M}.$
• ${\mathtt{A}}_n$ is in $\mathtt{M},n=1,2,\dots ,$ implies ${\displaystyle \bigcup _{n=1}^{\infty }}{\mathtt{A}}_n\in \mathtt{M}.$

Let $\mathcal{C}$ be an arbitrary collection of subsets of $\mathtt{X}.$ The smallest $\sigma$-algebra $\sigma (\mathtt{C})$ containing $\mathtt{C},$ called the $\sigma$-algebra generated by $\mathtt{C},$ is the intersection of all $\sigma$-algebras in $\mathtt{X}$ which contain $\mathtt{C}.$

Let $\mathtt{M}$ be a $\sigma$-algebra of subsets of a set $\mathtt{X}.$ A positive function $\mu :\mathtt{M}\to [0,+\infty ]$ is called a measure if

• $\mu (\varnothing )=0;$
• $\mu ({\displaystyle \bigcup _{j=1}^{\infty }}{\mathtt{A}}_j)={\displaystyle \sum _{j=1}^{\infty }}\mu ({\mathtt{A}}_j)$ for every sequence ${\left\{{\mathtt{A}}_j\right\}}_j$ of pointwise disjoint sets from $\mathtt{M}.$

Then, $(\mathtt{X},\mathtt{M},\mu )$ is termed as a measure space. Suppose U is the Borel $\sigma$-algebra of $\mathtt{X}.$ Recall that a measure $\mu$ defined on U is called locally finite if for every $x\in \mathtt{X}$, there is $r>0$ such that $\mu (B(x,r))<\infty ,$ where $B(x,r)$ is the open ball of center x and radius $r.$ $\mu$ is called a Radon measure if $\mu$ is a Borel measure with the followings:

• $\mu (\mathtt{K})<\infty$ for every compact set $\mathtt{K}\subset \mathtt{X}.$
• $\mu (\mathtt{V})=sup\{\mu (\mathtt{K}):\mathtt{K}\subset \mathtt{V},\mathtt{K}\mathrm{is}\mathrm{compact}\}$ for every open set $\mathtt{V}\subset \mathtt{X}.$
• $\mu (\mathtt{A})=inf\{\mu (\mathtt{V}):\mathtt{A}\subset \mathtt{V},\mathtt{V}\mathrm{is} \mathrm{open}\}$ for every $\mathtt{A}\subset \mathtt{X}.$

In the entire work, we consider $\mu$ a non-atomic Radon measure, $\mathtt{F}$ is a family of $\mu$-measurable sets in short $\mu$ sets and $Q\in \mathtt{F}.$ For $\mathtt{E}\subset \mathtt{X},$ we denote the indicator function, diameter, interior and the boundary of $\mathtt{E}$ by ${\chi }_{\mathtt{E}},\mathrm{diam}(\mathtt{E}),{\mathtt{E}}^0$ and $\partial \mathtt{E},$ respectively. Throughout the article, we denote $\mathfrak{d}(x,\mathtt{E})$ as the distance from x to $\mathtt{E}.$ Let us define $\mathtt{F}$ as a family of non-empty closed subsets of $\mathtt{X}.$

If ${\displaystyle \bigcup _{i=1}^m}Qi=Q,$ then a partition of Q is a finite gathering of $Q_1,Q_2,\dots ,Q_n$ pairwise non-overlapping elements of $\mathtt{F}$.

Let $\mathtt{E}\subset \mathtt{X}$ and let $\mathfrak{F}$ be a subfamily of $\mathtt{F}.$ We say that $\mathfrak{F}$ is a fine cover of $\mathtt{E}$ if

$$\begin{array}{c}\hfill inf\left\{\mathrm{diam}(Q):Q\in \mathfrak{F},x\in Q\right\}=0\end{array}$$

for each $x\in \mathtt{E}.$

Definition 1 

([13], Definition 2.14). We say that $\mathtt{F}$ is a μ-Vitali family if for each subset $\mathtt{E}$ of $\mathtt{X}$ and for each subfamily $\mathfrak{F}$ of $\mathtt{F}$ that is a fine cover of $\mathtt{E},$ there exists a countable system $\left\{Q_1,Q_2,..,Q_j,..\right\}$ of pairwise non-overlapping elements of $\mathfrak{F}$ such that $\mu (\mathtt{E}\backslash \bigcup Qj)=0.$

Consider a fine cover $\mathfrak{F}$ of $\mathtt{E}\subset \mathtt{X}.$ Recall that a family $\mathtt{F}$ of non-void closed subsets of $\mathtt{X}$ is a $\mu$ Vitali family if the following Vitali covering theorem is fulfilled:

Theorem 1 

([13], Theorem 2.1). For each subset $\mathtt{E}$ of $\mathtt{X}$ and for each subfamily $\mathfrak{F}$ of $\mathtt{F}$ that is a fine cover of $\mathtt{E},$ there exists a countable system $\left\{Q_1,Q_2,\dots ,Q_j,\dots \right\}\subset \mathfrak{F}$ such that $Q_i$ and $Q_j$ are non-overlapping (i.e., the interiors of $Q_i$ and $Q_j$ are disjoint), for each $i\ne j,$ and such that $\mu (\mathtt{E}\backslash \cup Q_j)=0.$

A $\mu$-Vitali family $\mathtt{F}$ is said to be a family of $\mu$ sets if it satisfies the following conditions:

(a)

Given $Q\in \mathtt{F}$ and a constant $\delta >0,$ there exist $Q_1,Q_2,\dots ,Q_m,\dots$ subsets of $Q,$ such that $Q_i$ and $Q_j$ are non-overlapping for each $i\ne j,{\displaystyle \bigcup _{i=1}^m}{Q}_i=Q,$ and $\mathrm{diam}(Q_i)<\delta ,$ for $i=1,2,\dots ,m;$

(b)

Given $A,Q\in \mathtt{F}$ with $A\subset Q,$ there exist $Q_1,Q_2,\dots ,Q_m,\dots$ subsets of $Q,$ such that $Q_i$ and $Q_j$ are non-overlapping for each $i\ne j$ and $A=Q_1;$

(c)

$\mu (\partial Q)=0$ for each $Q\in \mathtt{F}.$

The Vitali covering theorem is one of the most useful tools of measure theory. Given a collection of sets that cover some set $A,$ the Vitali theorem selects a disjoint subcollection that covers almost all of $A.$ Here, we recall Vitali’s covering theorem for Radon measures as follows:

Theorem 2 

([14], Page 34). Let μ be a Radon measure on ${\mathtt{R}}^n,A\subset {\mathtt{R}}^n$ and $\mathtt{B}$ a family of closed balls such that each point of A is the center of arbitrarily small balls of $\mathtt{B},$ that is

$$inf\left\{r:\mathtt{B}(x,r)\in \mathtt{B}\right\}=0,$$

for each $x\in A.$ Then, there are disjoint balls ${\mathtt{B}}_i\in \mathtt{B},i=1,2,\dots ,$ such that $\mu \left(A\backslash {\displaystyle \bigcup _i}{\mathtt{B}}_i\right)=0.$

Definition 2 

([15]). Consider a measurable set $\mathtt{E}$ included in $\mathtt{R}$ and c is a real number. At c, the density of $\mathtt{E}$ equals

$$\begin{array}{c}\hfill d_c(\mathtt{E})={\displaystyle \underset{h\to 0+}{lim}}\frac{\mu (\mathtt{E}\cap (c-h,c+h))}{2h}\end{array}$$

provided the limit exists. Clearly, $0\le d_c(\mathtt{E})\le 1$ when it exists. The point c is a point of density of $\mathtt{E}$ if $d_c\mathtt{E}=1$ and a point of dispersion of $\mathtt{E}$ if $d_c(\mathtt{E})=0.$

An approximate neighborhood of $x\in [a,b]$ is a measurable set $N_x\subset [a,b]$ containing x as a point of density. Let $\mathtt{E}\subset [a,b].$ For every $x\in \mathtt{E}\subset [a,b],$ choose an approximate neighborhood $N_x\subset [a,b]$ of $x.$ Then, $N=\left\{N_x:x\in E\right\}$ is a choice on $\mathtt{E}.$ If each point of $N_x$ is a point of density of $N_x$, then a tagged interval $([c,d],x)$ is said to be fine to the choice $N=\left\{N_x\right\}$ if $c,d\in N_x$ and $x\in [c,d].$ A tagged subpartition $P=\left\{([c_i,d_i],t_i):1\le i\le n\right\}$ of $[a,b]$ is a finite collection of non-overlapping tagged intervals in $[a,b]$ such that $t_i\in [c_i,d_i]$ for $i=1,2,\dots ,n.$ If $([c_i,d_i],t_i)$ is fine to the choice N for each $i=1,2,\dots ,n;$ then, P is N-fine. If P is N-fine and $t_i\in \mathtt{E}$ for each $1\le i\le n,$ then we say that P is $(N,\mathtt{E})$-fine. If P is N-fine and $t_i\in \mathtt{E}$ for each $1\le i\le n,$ then P is $(N,\mathtt{E})$-fine. If P is N-fine and $[a,b]={\displaystyle {\displaystyle \bigcup _{i=1}^n}}[c_i,d_i],$, then P is an N-fine tagged partition of $[a,b].$ For a tagged partition $P=\left\{(x_i,[c_i,d_i]):1\le i\le n\right\}$ of $[a,b],$ the Riemann sum is $\mathbb{S}(f,P)={\displaystyle {\displaystyle \sum _{i=1}^n}}f(x_i)(d_i-c_i).$

Consider a metric measure space $(\mathtt{X},\mathfrak{d},\mu )$ in the sense that the metric induces a topology $\mathfrak{T},$ and the measure is the Borel measure arising from the sigma field induced by the metric $\mathfrak{d}.$ If $\mathfrak{T}$ is second countable locally compact and Hausdorff, we can consider a basis consisting of relatively compact open sets. Let ${\mathfrak{T}}_1$ be a basis for $\mathfrak{T}$ consisting of relatively compact open sets. If the topology $\mathfrak{T}$ of $\mathtt{X}$ is induced by a metric $\mathfrak{d}$ on $\mathtt{X}$, then ${\mathfrak{T}}_1$ is the set of all $\mathfrak{d}$-open balls say $\mathtt{B}(x,r)$ are $\mathfrak{d}-$open balls with $\mu (\mathtt{B}(x,r))<ϵ.$

Let $\mathfrak{u}\in {\mathfrak{T}}_1$ and $\mathfrak{u}\ne \varnothing$; then, $\mu (\mathfrak{u})>0.$ In addition, if $\overline{\mathfrak{u}}$ is a closure of $\mathfrak{u}$, then $\mu (\mathfrak{u})=\mu (\overline{\mathfrak{u}}).$ This means $\mu (\partial \mathfrak{u})=0$ for all $\mathfrak{u}\in {\mathfrak{T}}_1.$ Consider

$$\begin{array}{c}\hfill Q_0=\left\{\overline{{\mathfrak{u}}_1}\backslash \overline{{\mathfrak{u}}_2},{\mathfrak{u}}_1,{\mathfrak{u}}_2\in {\mathfrak{T}}_1\mathrm{with}{\mathfrak{u}}_1⊈{\mathfrak{u}}_2\mathrm{and}{\mathfrak{u}}_2⊈{\mathfrak{u}}_1\right\}\end{array}$$

and

$$\begin{array}{c}\hfill Q_1=\left\{{\displaystyle \bigcap _{i\in \mathbb{N}}}{\mathfrak{v}}_i\ne \varnothing ;{\mathfrak{v}}_i\in Q_0\right\},\end{array}$$

then $Q_0$ includes all sets of the form $\overline{\mathfrak{u}}$ where $\mathfrak{u}\in {\mathfrak{T}}_1$ and $Q_0\subseteq Q_1.$ Clearly, $Q_1$ is closed under finite intersections. If the intersection is non-empty and since $\mu$ is a $\sigma$-algebra, members of $Q_1$ are $\mu$-measurable. A set Q is called elementary if Q is a finite union of (possibly just one) mutually disjoint sets. We say that a set $P=\{(Q_i,x_i):i=1,2,\dots ,n\}$ is a partial partition of Q if $Q_1,Q_2,\dots ,Q_n$ are mutually disjoint subsets of Q such that $Q\backslash {\bigcup }_{i=1}^n{Q}_i=\varnothing$ or elementary subset of Q for each $i,$ with $x_i\in \overline{Q_i}.$ Throughout the article, the closure of Q is denoted by $\overline{Q}.$

Let $\mathrm{\Gamma }:\overline{Q}\to {\mathfrak{T}}_1$ be a function such that for every $x\in \overline{Q},$ we have $x\in \mathrm{\Gamma }(x)\in {\mathfrak{T}}_1.$ We call $\mathrm{\Gamma }$ as a gauge on $Q.$ If $\mathfrak{T}$ of $\mathtt{X}$ is induced by the metric $\mathfrak{d},$ then ${\mathfrak{T}}_1,$ the collection of all $\mathfrak{d}$-open balls and a gauge $\mathrm{\Gamma }$ on Q be as $\mathrm{\Gamma }(x)=\mathcal{B}(x,\delta (x))$ for all $x\in \overline{Q}$ with certain $\delta (x)>0.$ If $\mathrm{\Gamma }$ is a gauge on $Q,$ then $(Q_i,x_i)$ is N-fine if $Q\subseteq \mathrm{\Gamma }(x).$ If for all $i=1,2,\dots ,n,$ $(Q_i,x_i)$ is N-fine, then $P=\{(Q_i,x_i):i=1,2,\dots ,n\}$ of Q is also N-fine. Given that partitions of Q are only partial partitions of $Q,$ an N-fine partition of Q can be defined similarly. Let ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$ be two gauges on $Q.$ Since ${\mathfrak{T}}_1$ is a basis of $\mathfrak{T},$ for each $x\in Q,$ there exists $\mathrm{\Gamma }(x)\in {\mathfrak{T}}_1$ such that $\mathrm{\Gamma }(x)\subseteq {\mathrm{\Gamma }}_1(x)\cap {\mathrm{\Gamma }}_2(x).$ We can then define a gauge $\mathrm{\Gamma }$ on Q which is finer than both ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2.$ As a result, P is both ${\mathrm{\Gamma }}_1$-fine and ${\mathrm{\Gamma }}_2$-fine if P is a N-fine partition of $Q.$

Proposition 1 

([13], Lemma 2.2.1). (Cousin’s-type lemma) if δ is a gauge on $Q,$ then there exists a δ-fine partition of $Q.$

Recalling the $AP$–Henstock–Kurzweil integral as follows:

Definition 3 

([16], Definition 16.4). A mapping $\mathtt{f}:[a,b]\to \mathtt{R}$ is called an $AP$–Henstock–Kurzweil integrable if a real number A exists such that for each $ϵ>0$, there is a choice N on $[a,b]$ such that $|\mathbb{S}(f,P)-A|<ϵ$ whenever P is an N-fine tagged partition of $[a,b].$ In this case, A is called the $AP$–Henstock–Kurzweil integral of f on $[a,b],$ and is denoted by $A={\int }_a^{b}f.$

Given a set function $\mathbb{F}$ defined on $\mathtt{F}$ and given $x\in \mathtt{X},$ the upper derivative of $\mathbb{F}$ at $x,$ with respect to $\mu ,$ is defined as

$$\begin{array}{c}\hfill UD\mathbb{F}(x)={\displaystyle \underset{\mathtt{B}\to x\in \mathtt{F}}{lim}}sup\frac{\mathbb{F}(\mathtt{B})}{\mu (\mathtt{B})}.\end{array}$$

Here, $\mathtt{B}\to x$ implies $\mu (\mathtt{B})\ne 0,\mathrm{diam}(\mathtt{B})\to 0$ and $x\in \mathtt{B}.$

The lower derivative $LD\mathbb{F}$ is defined similarly. $UD\mathbb{F}$ and $LD\mathbb{F}$ are studied in [12].

Let $\varphi :\mathtt{F}\to \mathtt{R}$ be a function. We say that $\varphi$ is an additive function of set if for each $Q_i\in \mathtt{F},$ for all $i=1,2,\dots ,m$ and for each division $\{Q_1,Q_2,\dots ,Q_m\}$ of Q, we have

$$\begin{array}{c}\hfill \varphi (Q)={\displaystyle \sum _{i=1}^m}\varphi (Q_i).\end{array}$$

Proposition 2 

([17], page 5). For every measurable set $\mathtt{W}$ and every $ϵ>0,$ there exist an open set $\mathtt{U}$ and a closed set $\mathtt{Y}$ such that $\mathtt{Y}\subseteq \mathtt{W}\subseteq \mathtt{U}$ and $\mu (\mathtt{U}\backslash \mathtt{Y})<ϵ.$

3. AP–Henstock–Kurzweil Integral in Regard to a Radon Measure

In this section, we discuss the AP–Henstock–Kurzweil integral with respect to a Radon measure. We consider $\mu$ a non-atomic Radon measure and $Q\in \mathtt{F},$ where $\mathtt{F}$ is a family of $\mu$ sets. An approximate neighbourhood $x\in Q$ is a measurable set $N_x\subset Q$ containing x as a point of density. Suppose $\mathtt{E}\subset Q.$ For every $x\in \mathtt{E}\subset Q,$ choose an approximate neighborhood $N_x\subset Q$ of $x.$ Then, $N=\{N_x:x\in \mathtt{E}\}$ is a choice on $\mathtt{E}.$ If each point of $N_x$ is a point of density of $N_x$, then a tagged $({\mathtt{E}}_i,x)$ is said to be fine to the choice $N=\left\{N_x\right\}$ if $E_i\in N_x$ and $x\in E_i.$

Definition 4 

([15]). Given a μ-measurable set $\mathtt{E}\subset Q,$ a set-valued function $N:\mathtt{E}\to 2^Q$ is called an AP-neighborhood function (ANF) on $\mathtt{E}$ if for every $x\in \mathtt{E},$ there exists an $ap$-neighbourhood $N_x\subset Q$ of x such that $N(x)=N_x.$

A tagged subpartition, denoted by the symbol $P=\{({\mathtt{E}}_i,t_i):1\le i\le n\},$ consists of a finite set of non-overlapping tagged subsets in Q that way $t_i\in {\mathtt{E}}_i$ for $i=1,2,3,\dots ,n.$ We say that P is N-fine if $({\mathtt{E}}_i,t_i)$ is acceptable to the selection N for $i=1,2,\dots ,n$. If P is N-fine and $t_i\in \mathtt{E}$ for each $1\le i\le n,$ we may say that P is $(N,\mathtt{E})$ fine. We refer to P as the tagged partition of Q if P is N-fine and $Q={\displaystyle {\displaystyle \bigcup _{i=1}^n}}{\mathtt{E}}_i.$

Definition 5. 

Let $\mathtt{f}$ be a function defined on μ set Q. We say $\mathtt{f}:Q\subset \mathtt{F}\to \mathtt{R}$ is approximately continuous at $c\in Q$ if there exist a measurable set $\mathtt{E}\subset Q$ with density1 at c such that

$${\displaystyle \underset{x\to c}{lim}}\mathtt{f}(x)=Aforx\in \mathtt{E}.$$

We say $\mathtt{f}$ is approximately differentiable at c if there exists a real number A and a measurable set $\mathtt{E}\subset Q$ such that the density of $\mathtt{E}$ at c is 1 and

$${\displaystyle \underset{x\to c}{lim}}\frac{\mathtt{f}(x)-\mathtt{f}(c)}{x-c}=Aforx\in \mathtt{E}.$$

We denote $A={\mathtt{f}}_{ap}^{\prime }(c).$ For a tagged partition $P=\{({\mathtt{E}}_i,t_i):1\le i\le n\}$ of $Q,$ we denote $\mathbb{S}(\mathtt{f},P)={\displaystyle \sum _{i=1}^n}\mathtt{f}(t_i)\mu ({\mathtt{E}}_i)$ and $\mathtt{f}(P)=\mathbb{S}(\mathtt{f},P).$ For a function $F_1:Q\to \mathtt{R},F_1$ can be utilized as a set-based function by specifying $F_1(Q)=\mu (Q).$ We will denote $F_1(P)={\displaystyle \sum _{i=1}^n}{F}_1(Q_i)$ for an N-fine tagged partition $P=\{(Q_i,t_i),1\le i\le n\}.$

Definition 6. 

A function $\mathtt{f}:Q\to \mathtt{R}$ is called ${\mu }_{AP}$–Henstock–Kurzweil integrable if there exists a real number A such that for each $ϵ>0$, there exists a choice N on Q such that $|\mathbb{S}(\mathtt{f},P)-A|<ϵ.$

In this instance, A is referred to as the ${\mu }_{AP}$–Henstock–Kurzweil integral of $\mathtt{f}$ on $Q,$ and we write

$$\begin{array}{c}\hfill A=(AP){\int }_Q\mathtt{f}d\mu .\end{array}$$

(1)

The collection of all integrable ${\mu }_{AP}$–Henstock–Kurzweil functions on Q (in regard to $\mu$) shall be written as ${\mu }_{AP}HK(Q).$ It is observe that A of (1) is unique. The ${\mu }_{AP}HK$ integral includes the $AP$-real line Henstock–Kurzweil integral.

Example 1. 

Using Euclidean distance in $\mathtt{R}$ along with the Lebesgue measure $\lambda ,$ let $\mathtt{X}$ be the interval of the real line $[0,1]$. If we consider

$$\mathtt{F}=\{I:Iisnonemptyclosedsubintervalof\mathtt{X}\}$$

Clearly, $\mathtt{F}$ is a μ set in $[0,1]$ and the ${\mu }_{AP}$–Henstock–Kurzweil integral is the usual $AP$–Henstock–Kurzweil integral on $[0,1].$

Remark 1. 

Since the intersection of two approximate full covers of Q is another approximate full cover of $Q,$ also the number A of (1) is unique. Every μ–Henstock–Kurzweil integrable function is certainly ${\mu }_{AP}$–Henstock–Kurzweil integrable and the integrals are equal.

3.1. Simple Properties

Here, we will study some fundamental characteristics of ${\mu }_{AP}HK(Q)$ integrable functions.

Theorem 3. 

1. Suppose ${\mathtt{f}}_1,{\mathtt{f}}_2\in {\mu }_{AP}HK(Q)$ on this occasion ${\mathtt{f}}_1+{\mathtt{f}}_2\in {\mu }_{AP}HK(Q)$ along with

$$(AP){\int }_Q({\mathtt{f}}_1+{\mathtt{f}}_2)d\mu =(AP){\int }_Q{\mathtt{f}}_{1}d\mu +(AP){\int }_Q{\mathtt{f}}_{2}d\mu .$$

2. Suppose ${\mathtt{f}}_1\in {\mu }_{AP}HK(Q)\alpha$ is a scalar, then $\alpha {\mathtt{f}}_1\in {\mu }_{AP}HK(Q)$ with $(AP){\int }_Q\alpha {\mathtt{f}}_{1}d\mu =(AP)\alpha {\int }_Q{\mathtt{f}}_{1}d\mu .$

Proof. 

For $(1)$ Let $ϵ>0$ be given and suppose $A_1,A_2$ are ${\mu }_{AP}$–Henstock–Kurzweil integrals of ${\mathtt{f}}_1,{\mathtt{f}}_2$, respectively. Since ${\mathtt{f}}_1\in {\mu }_{AP}HK(Q),$ consider a gauge ${\delta }_1$ on Q such that

$$|\mathbb{S}({\mathtt{f}}_1,P_1)-A_1|<\frac{ϵ}{2},$$

for each ${\delta }_1$-fine partition $P_1$ of $Q.$ Similarly, there exists a positive function (gauge) ${\delta }_2$ on Q so that for every ${\delta }_2$-fine partition $P_2$ of Q, we have

$$|\mathbb{S}({\mathtt{f}}_2,P_2)-A_2|<\frac{ϵ}{2}.$$

Assuming a gauge on $Q,$ with $\delta =min\{{\delta }_1,{\delta }_2\}.$ Being aware of the fact that if $\delta$ is a gauge on $Q,$ then there is a $\delta$-fine partition of $Q.$ Since P is both ${\delta }_1,{\delta }_2$-fine, we can find

$$|\mathbb{S}({\mathtt{f}}_1+{\mathtt{f}}_2,P)-(A_1+A_2)|<ϵ.$$

Therefore, ${\mathtt{f}}_1+{\mathtt{f}}_2\in {\mu }_{AP}HK(Q).$

The proof of $(2)$ is similar to the proof of $(1).$ □

Theorem 4. 

Suppose $\mathtt{f}\in {\mu }_{AP}HK(Q)$ and suppose for every $x\in Q,\mathtt{f}(x)\ge 0.$ This will give $(AP){\int }_Q\mathtt{f}d\mu \ge 0.$

Proof. 

Let $ϵ>0$ be given. Since $\mathtt{f}\in {\mu }_{AP}HK(Q),$ there exists a gauge $\delta$ on Q such that

$$\left|\mathbb{S}(\mathtt{f},P)-(AP){\int }_Q\mathtt{f}d\mu \right|<ϵ,$$

for each $\delta$-fine partition $P={\left\{(x_i,Q_i)\right\}}_{i=1}^m$ of $Q.$ Since $\mathtt{f}(x)\ge 0$ for each $x\in Q,$ we have

$$\mathbb{S}(\mathtt{f},P)={\displaystyle \sum _{i=1}^m}\mathtt{f}(x_i)\mu (Q_i)\ge 0.$$

Therefore

$$\begin{array}{c}\hfill \mathbb{S}(\mathtt{f},P)-ϵ<(AP){\int }_Q\mathtt{f}d\mu <\mathbb{S}(\mathtt{f},P)+ϵ.\end{array}$$

Since $ϵ$ is arbitrary, we obtain $(AP){\int }_Q\mathtt{f}d\mu \ge 0.$ □

Theorem 5. 

Suppose ${\mathtt{f}}_1,{\mathtt{f}}_2\in {\mu }_{AP}HK(Q).$ In addition, if ${\mathtt{f}}_1\le {\mathtt{f}}_2$ for every $x\in Q,$ then $(AP){\int }_Q{\mathtt{f}}_{1}d\mu \le (AP){\int }_Q{\mathtt{f}}_{2}d\mu .$

Proof. 

Let $\mathtt{f}={\mathtt{f}}_2-{\mathtt{f}}_1.$ By Theorem 3, we have $\mathtt{f}\in {\mu }_{AP}HK(Q)$ and

$$\begin{array}{c}\hfill (AP){\int }_Q\mathtt{f}d\mu =(AP){\int }_Q{\mathtt{f}}_{2}d\mu -(AP){\int }_Q{\mathtt{f}}_{\mathtt{1}}d\mu .\end{array}$$

Since ${\mathtt{f}}_1\le {\mathtt{f}}_2$ then $\mathtt{f}\ge 0$ for each $x\in Q$ and by Theorem 4, we obtain that $(AP){\int }_Q\mathtt{f}d\mu \ge 0.$ Therefore $(AP){\int }_Q{\mathtt{f}}_{1}d\mu \le (AP){\int }_Q{\mathtt{f}}_{2}d\mu .$ □

Theorem 6 

(The Cauchy Criterion). A mapping $\mathtt{f}:Q\to \mathtt{R}$ is ${\mu }_{AP}$HK integrable on Q if and only if for every $ϵ>0$, there exists a positive function (gauge) δ on Q such that

$$|\mathbb{S}(\mathtt{f},P_1)-\mathbb{S}(\mathtt{f},P_2)|<ϵ,$$

for each pair N-fine partitions $P_1$ and $P_2$ of $Q.$

Proof. 

Let us consider $\mathtt{f}:Q\to \mathbb{R}$ is ${\mu }_{AP}$HK integable on $Q.$ Given $ϵ>0,$ there exists a gauge $\delta$ on Q such that

$$\left|\mathbb{S}(\mathtt{f},P)-(AP){\int }_Q\mathtt{f}d\mu \right|<\frac{ϵ}{2},$$

for each N-fine partition P of $Q.$ If $P_1$ and $P_2$ are two N-fine partitions of $Q,$ we have

$$\begin{array}{cc}\hfill |\mathbb{S}(\mathtt{f},P_1)-\mathbb{S}(\mathtt{f},P_2)|& \le \left|\mathbb{S}(\mathtt{f},P_1)-(AP){\int }_Q\mathtt{f}d\mu \right|+\left|\mathbb{S}(\mathtt{f},P_2)-(AP){\int }_Q\mathtt{f}d\mu \right|\hfill \\ & <ϵ.\hfill \end{array}$$

Conversely, for each $n\in \mathbb{N}$, let ${\delta }_n$ be a gauge on Q such that

$$|\mathbb{S}(\mathtt{f},P_n^{{}^{\prime }})-\mathbb{S}(\mathtt{f},P_n^{{}^{″}})|<\frac{1}{n}$$

for each pair N-fine partitions $P_n^{{}^{\prime }}$ and $P_n^{{}^{″}}$ of $Q.$ Let ${\mathsf{\delta }}_n(x)=min\{{\delta }_1(x),{\delta }_2(x),\dots ,{\delta }_n(x)\}$ be a gauge on $Q.$ By the Proposition 1, there exists a $\mathsf{\delta }$-fine partition (respectively, N-fine) of $P_n$ of $Q,$ for each $n\in \mathbb{N}.$ Let $ϵ>0$ be given and choose a positive natural N such that $\frac{1}{N}<\frac{ϵ}{2}.$ If m and n are positive natural ($n<m$) such that $n\ge N,$ then $P_n$ and $P_m$ are N-fine partitions on $Q;$ hence

$$|\mathbb{S}(\mathtt{f},P_n)-\mathbb{S}(\mathtt{f},P_m)|<\frac{1}{n}<\frac{ϵ}{2}.$$

Consequently, ${\left\{\mathbb{S}(\mathtt{f},P_n)\right\}}_{n=1}^{\infty }$ is a Cauchy sequence of real numbers and hence converges.

If $A={\displaystyle \underset{n\to \infty }{lim}}\mathbb{S}(\mathtt{f},P_n),$ then

$$|\mathbb{S}(\mathtt{f},P_n)-A|<\frac{ϵ}{2},$$

for each $n\ge N.$ Let P be an N-fine partition on $Q,$ then

$$\begin{array}{cc}\hfill |\mathbb{S}(\mathtt{f},P)-A|& \le |\mathbb{S}(\mathtt{f},P)-\mathbb{S}(\mathtt{f},P_N)|+|\mathbb{S}(\mathtt{f},P_N)-A|\hfill \\ & <ϵ.\hfill \end{array}$$

Hence, f is ${\mu }_{AP}$HK integrable on $Q.$ □

Corollary 1. 

Corollary Suppose $Q\in \mathtt{F}$ and $\mathtt{f}\in {\mu }_{AP}HK(Q).$ Assume there exists a partition D of Q with $A\in D.$ Then, $\mathtt{f}$ is in ${\mu }_{AP}HK(A)$ with $(AP){\int }_A\mathtt{f}d\mu =(AP){\int }_Q\mathtt{f}{\chi }_{A}d\mu .$

Remark 2. 

1. Let $\mathtt{f}:Q\to \mathtt{R}$ be ${\mu }_{AP}$–Henstock–Kurzweil integrable on $Q;$ then, $\mathtt{f}$ is ${\mu }_{AP}$–Henstock–Kurzweil integrable on every subset $E_i\subset Q\in \mathtt{F}.$

2. If $\mathtt{f}$ is ${\mu }_{AP}$–Henstock–Kurzweil integrable on each subset $E_i\subset Q$, then $\mathtt{f}$ is ${\mu }_{AP}$–Henstock–Kurzweil integrable on $Q.$

Proposition 3. 

Consider $\mathtt{f}:Q\to \mathtt{R}$ in ${\mu }_{AP}(HK)(Q)$ and a partition $\{Q_1,Q_2,\dots ,Q_m\}$ of $Q.$ Then

$$\begin{array}{c}\hfill \mathtt{f}\in {\mu }_{AP}(HK)(Q_1)\cap {\mu }_{AP}(HK)(Q_2)\cap ..\cap {\mu }_{AP}(HK)(Q_m),\end{array}$$

with $(AP){\int }_Q\mathtt{f}d\mu ={\displaystyle \sum _{i=1}^m}(AP){\int }_{Q_i}\mathtt{f}d\mu .$

Proof. 

Since $\mathtt{f}:Q\to \mathtt{R}$ is ${\mu }_{AP}$–HK integrable on $Q,$ for given $ϵ>0$ and for a gauge $\delta ,$ there will be a choice N on Q so that

$$\begin{array}{c}\hfill \left|\mathbb{S}(\mathtt{f},P)-(AP){\int }_Q\mathtt{f}d\mu \right|<ϵ.\end{array}$$

Now, according to Corollary 1, $\mathtt{f}$ is in ${\mu }_{AP}(HK)(Q_i)$ for $i=1,2,\dots ;m$; then, there exists a choice $N_i$ for gauge ${\delta }_i$ on $Q_i$ for $i=1,2,\dots ,m$ such that ${\delta }_i(x)<\delta (x)$ for each x, and

$$\begin{array}{c}\hfill \left|\mathbb{S}(\mathtt{f},P_i)-(AP){\int }_{Q_i}\mathtt{f}d\mu \right|<\frac{ϵ}{m},\end{array}$$

for each ${\delta }_i$-fine partition $P_i$ of $Q.$ If $\delta =max\{{\delta }_i:1,2,\dots \}$, then $P=P_1\cup P_2\cup \dots \cup P_m$ is a $\delta$-fine partition on Q for the choice $max\left\{N_i\right\}=N.$ Consequently,

$$\begin{array}{cc}\hfill \left|\mathbb{S}(\mathtt{f},P)-{\displaystyle \sum _{i=1}^m}(AP){\int }_{Q_i}\mathtt{f}d\mu \right|& \le \left|\mathbb{S}(\mathtt{f},P_1)-(AP){\int }_{Q_1}\mathtt{f}d\mu \right|+\dots +\left|\mathbb{S}(\mathtt{f},P_m)-(AP){\int }_{Q_m}\mathtt{f}d\mu \right|\hfill \\ & <ϵ.\hfill \end{array}$$

So, $(AP){\int }_Q\mathtt{f}d\mu ={\displaystyle \sum _{i=1}^m}(AP){\int }_{Q_i}\mathtt{f}d\mu .$ □

Definition 7. 

We call the map $\overline{F}:\mathtt{E}⇝(AP){\int }_{\mathtt{E}}fd\mu$ defined on each subset $\mathtt{E}$ of Q the indefinite ${\mu }_{AP}HK$ integral of $\mathtt{f}$ on $Q.$

Theorem 7. 

The map $\overline{F}\to \mathtt{E}⇝(AP){\int }_{\mathtt{E}}\mathtt{f}d\mu$ of Definition 7 is an additive set function.

Proof. 

The proof follows from Proposition 3. □

Definition 8. 

A collection S of N-tagged in $\mathfrak{F}$ is an approximate full cover of $Q\subset \mathfrak{F}$ if for each $x\in Q$ there exists a measurable set $S_x\subset \mathfrak{F}$ such that for a compact subset $\mathtt{E},x\in S_x$ and $(x,\mathtt{E})\subset S$ if and only if $x_{\mathtt{E}}\in \mathtt{E}\in S_x.$ The collection $S=\{S_x:x\in Q\}$ is called the collection of sets generated by $S.$ If $Q_1\subset Q$, then $S_x=\{(x,\mathtt{E})\in S:x\in Q_1\}.$

Theorem 8. 

Let ϕ be a set function that is defined on the class of all subsets of $Q.$ Suppose ϕ is additive. A mapping $\mathtt{f}:Q\to \mathtt{R}$ is ${\mu }_{AP}$HK integrable on Q if and only if for each $ϵ>0$, there exists an N-fine gauge δ on Q with

$$\begin{array}{c}\hfill {\displaystyle \sum _{(x_i,Q_i)\in P}}|\varphi (Q_i)-\mathtt{f}(x_i)\mu (Q_i)|<ϵ\end{array}$$

for every P partial partition of Q that is N-fine.

Proof. 

Proof is similar to the ([13], Lemma 2.4.1), so we omit the proof. □

Now, we discuss the properties of the ${\mu }_{AP}$–Henstock–Kurzweil integral with approximate differentiation with respect to the Radon measure $\mu .$ Recall that a Radon measure $\mu$ is a Borel regular measure if it is a Borel measure. Additionally, a Borel regular measure $\mu$ becomes a Borel measure if for each $\mathtt{E}\subset \mathtt{X}$, there exists a Borel subset B of $\mathtt{X}$ such that $\mathtt{E}\subset \mathtt{B}$ and $\mu (\mathtt{B})=\mu (\mathtt{E}).$ In addition, see [13,14] for details.

In order to consider the upper approximate differentiation at the point of density, with respect to $\mu$, we define the following notions:

Definition 9. 

Let us consider $\overline{F}$ a set function defined on $\mathtt{F}.$ For a given $x\in \mathtt{X},$ the upper approximate differentiation at x with regard to μ is determined by

$$\begin{array}{c}\hfill U_{AD}\overline{F}(x)=inf\left\{\alpha \in \mathtt{R}:xisapointofdispersionof\left\{x\in Q:{\displaystyle \underset{\mathtt{B}\to x\in \mathtt{F}}{lim}}\frac{\overline{F}(\mathtt{B})}{\mu (\mathtt{B})}\ge \alpha \right\}\right\},\end{array}$$

where $\mathtt{B}\to x$ means $\mu (\mathtt{B})\ne 0,diam(\mathtt{B})\to 0$ and $x\in \mathtt{B}.$

Similarly, we define the lower approximate derivative of $\overline{F}$ at x with respect to $\mu$ as

$$\begin{array}{c}\hfill L_{AD}\overline{F}(x)=sup\left\{\beta \in \mathtt{R}:\mathrm{x}\mathrm{is}\mathrm{a}\mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\left\{x\in Q:{\displaystyle \underset{\mathtt{B}\to x\in \mathtt{F}}{lim}}\frac{\overline{F}(\mathtt{B})}{\mu (\mathtt{B})}\le \beta \right\}\right\},\end{array}$$

where $\mathtt{B}\to x$ means $\mu (\mathtt{B})\ne 0,\mathrm{diam}(\mathtt{B})\to 0$ and $x\in \mathtt{B}.$

When $U_{AD}\overline{F}(c)$ and $L_{AD}\overline{F}(c)$ are equal but different from ∞ and $-\infty$, then $\overline{F}$ is called approximate differentiable at $c.$ The common value is known as the approximate derivative of $\overline{F}$ at $c.$ It is described as ${\overline{F}}_{AP}^{\prime }(x).$

Clearly, $LD\overline{F}(x)\le L_{AP}\overline{F}(x)\le U_{AP}\overline{F}(x)\le UD\overline{F}(x).$ The properties of the approximate derivatives are similar to those for ordinary derivatives with respect to $\mu .$

Theorem 9. 

1. If $\mathtt{f}$ is a non-negative ${\mu }_{AP}HK$ integrable function on a set Q and $\overline{F}$ is its indefinite ${\mu }_{AP}HK$-integral, then $\overline{F}$ is approximate differentiable μ-almost everywhere on Q and ${\overline{F}}_{AP}^{{}^{\prime }}=f\mu$-a.e.

2. The function $\overline{F}$ is μ-measurable.

Proof. 

We prove $(1)$ by contradiction. Suppose ${\overline{F}}_{AP}^{\prime }\ne \mathtt{f}$ a.e on $Q,$ then at least one of the sets

$$\begin{array}{c}\hfill \left\{x\in Q:\mathtt{f}(x)-L_{AD}\overline{F}(x)>0\right\}\end{array}$$

(2)

$$\begin{array}{c}\hfill \left\{x\in Q:-\mathtt{f}(x)+L_{AD}\overline{F}(x)>0\right\}\end{array}$$

(3)

has a positive outer measure. Due to the positive outer measure of Equation (2), there exists positive numbers $r_0,r_1$ so that ${\mu }^{*}(A)>r_0$ where $A=\left\{x\in Q:\mathtt{f}(x)-L_{AD}\overline{F}(x)>r_1\right\}.$ Let $\mathtt{S}$ be the system of all Borel sets $\mathtt{B}\subset Q$ such that $\overline{F}(\mathtt{B})>\mu (\mathtt{B})$ and there exists $x\in \mathtt{B}\cap A$ with $\mathrm{diam}(\mathtt{B})<\delta (x).$ It is easy to see that $\mathtt{S}$ is a fine cover of $A.$ Therefore, there exists a system of pairwise non-overlapping sets ${\left\{{\mathtt{B}}_i\right\}}_{i=1}^m\subset \mathtt{S}$ such that $\mu (A)\le {\displaystyle \sum _{i=1}^m}\mu ({\mathtt{B}}_i)+ϵ.$ Now, $\mathtt{f}$ being ${\mu }_{AP}$–Henstock–Kurzweil integrable on $Q,$ there exists an approximate full cover $\mathtt{S}$ of Q so that $|\mathtt{S}(\mathtt{f},P)-\overline{F}(P)|<r_0{r}_1,\forall P\subseteq \mathtt{S}.$ Let $\{{\mathtt{S}}_x:x\in Q\}$ be the collection of sets generated by $\mathtt{S}.$ Since $L_{AD}\overline{F}(x)<\mathtt{f}(x)-r_1$ for each $x\in A,$x is not a point of dispersion of the set

$${\mathtt{B}}_x=\left\{x\in Q:{\displaystyle \underset{\mathtt{B}\to x\in \mathtt{F}}{lim}}\frac{\overline{F}(\mathtt{B})}{\mu (\mathtt{B})}\le \mathtt{f}(x)-r_1\right\}.$$

Since x is a point of dispersion of $(x-\delta ,x+\delta )\backslash {\mathtt{S}}_x$ so, ${\mathtt{B}}_x\cap (x-\delta ,x+\delta )\cap {\mathtt{S}}_x\ne \varnothing$ for each $\delta >0.$ Let us choose a strictly monotone sequence $\left\{y_n^x\right\}\subseteq {\mathtt{B}}_x\cap {\mathtt{S}}_x\to x.$ Since ${\mu }^{*}(A)>r_0,$ so the collection $J={\displaystyle \bigcup _{x\in A}}\{[y_n^x,x]:n\in Z^{+}\}$ is a $\mu$-Vitali cover of $A.$ Now, by the Vitali Covering Lemma, there exists a finite collection $\{Q_i:1\le i\le q\}$ of disjoints subsets of J such that ${\displaystyle {\displaystyle \sum _{i=1}^q}}\mu (Q_i)>r_0.$ Let $P=\{(Q_i,x_i):1\le i\le q\}$ and $P\subset \mathtt{S}.$ Now,

$$\begin{array}{cc}\hfill \mathbb{S}(\mathtt{f},P)-\overline{F}(P)& ={\displaystyle \sum _{i=1}^q}\{\mathtt{f}(x_i)\mu ({\mathtt{Q}}_i)-\overline{F}(\mathtt{Q})\}\hfill \\ & ={\displaystyle \sum _{i=1}^q}\{\mathtt{f}(x_i)-\frac{\overline{F}(\mathtt{Q})}{\mu ({\mathtt{Q}}_i)}(\mu ({\mathtt{Q}}_i))\}\hfill \\ & \ge {\displaystyle \sum _{i=1}^q}{r}_1\mu ({\mathtt{Q}}_i)\hfill \\ & >r_0{r}_1\hfill \end{array}$$

which is a contradiction. So, ${\overline{F}}^{\prime }=f$ a.e. on $Q.$

For (2): To prove the function $\overline{F}$ is $\mu$-measurable, let $P_k$ be a $\frac{1}{k}N$-fine partial partition of Q and let ${\mathtt{f}}_k$ be the $\mu$-simple functions as

$${\mathtt{f}}_k(x)={\displaystyle \sum _{(x,\mathtt{B})\in P_k}}\frac{\overline{F}(\mathtt{B})}{\mu (\mathtt{B})},$$

where $\mathtt{B}$ is a Borel subset of $\mathtt{X}$ such that $\mathtt{E}\subset \mathtt{B}$ and $\mu (\mathtt{B})=\mu (\mathtt{E}).$ Let $\mathcal{C}={\displaystyle {\displaystyle \bigcup _{k=1}^{\infty }}}{\displaystyle {\displaystyle \bigcup _{\mathtt{B}\in P_k}}}\partial \mathtt{B}$ also when $D=\{x\in Q:{\overline{F}}_{AP}^{\prime }$ does not exist, or ${\overline{F}}_{AP}^{\prime }$ exists and ${\overline{F}}_{AP}^{\prime }(x)\ne \mathtt{f}(x)\}.$ Since $\mu (\partial \mathtt{B})=0$ for each $\mathtt{B}\in Q,$ and $\mathtt{f}$ is a non-negative ${\mu }_{AP}HK$ integrable function on a set Q with ${\overline{F}}_{AP}^{{}^{\prime }}=f$ $\mu$-a.e, we obtain that $Q_1=\mathcal{C}\cup D$ is $\mu$-null. Let $x\in Q\backslash Q_1.$ For each $k\in \mathbb{N}$, there exists $Q_{k,x}\in \mathtt{F}$ such that $(x,Q_{k,x})\in P_k,\mathrm{diam}(Q_k,x)<\frac{1}{k}$ and ${\mathtt{f}}_k(x)=\frac{\overline{F}(Q_{k,x})}{\mu (Q_k,x)}.$ Then, by ${\overline{F}}_{AP}^{\prime }=\mathtt{f}\mu$-a.e., we obtain ${\mathtt{f}}_k(x)\to \mathtt{f}(x).$ Therefore, $\overline{F}$ is $\mu$-measurable. □

Theorem 10. 

Every non-negative ${\mu }_{AP}$–Henstock–Kurzweil integrable function $\mathtt{f}$ on a set Q is μ-measurable if its indefinite integral $\overline{F}$ is ${\mu }_{AP}$–Henstock–Kurzweil integrable.

Proof. 

The proof is similar to the ([12], Theorem 5.3), so we omit the proof. □

Recall that on a set $Q,$ each Lebesgue integrable function coincides with the $\mu$–Henstock–Kurzweil integrable function. We recall the Vitali–Caratheódory Theorem below.

Theorem 11 

((Vitali–Caratheódory Theorem) [13], Theorem 2.5.1). Let $\mathtt{f}$ be a real function defined on a set $Q.$ If f is Lebesgue integrable on Q with respect to μ and $ϵ>0,$ then there exist functions $\mathtt{g}$ and $\mathtt{h}$ on Q such that $\mathtt{g}\le \mathtt{f}\le \mathtt{h},\mathtt{g}$ is upper semicontinuous and bounded above, $\mathtt{h}$ is lower semicontinuous and bounded below, and $(L){\int }_Q(\mathtt{h}-\mathtt{g})d\mu <ϵ.$

We find the relation between ${\mu }_{AP}$–Henstock–Kurzweil integrable functions and Lebesgue integrable functions on Q with the Lebesgue integral as follows:

Proposition 4. 

Every Lebesgue integrable function ${\mathtt{f}}_1:Q\to \mathtt{R}$ on Q with regard to μ is ${\mu }_{AP}$–Henstock–Kurzweil integrable on $Q.$ Consequently, $(L){\int }_Q{\mathtt{f}}_{1}d\mu =(AP){\int }_Q{\mathtt{f}}_{1}d\mu .$

Proof. 

Suppose ${\mathtt{f}}_1:Q\to \mathtt{R}$ is Lebesgue integrable on $Q.$ Using the Vitali–Caratheódory Theorem, for $ϵ>0$, there exist the functions ${\mathtt{f}}_2$ and ${\mathtt{f}}_3$ that are upper and lower semicontinuous, respectively, on Q such that $-\infty \le {\mathtt{f}}_2\le {\mathtt{f}}_1\le {\mathtt{f}}_3\le +\infty$ and $(L){\int }_Q({\mathtt{f}}_3-{\mathtt{f}}_2)d\mu <ϵ.$ Let $\delta$ be an N-fine gauge on Q so that ${\mathtt{f}}_2(t)\le {\mathtt{f}}_1(x)+ϵ$ and ${\mathtt{f}}_3(t)\ge {\mathtt{f}}_1(x)-ϵ$ for every $t\in Q$ along with $d(x,t)<\delta (x).$ For an N-fine partition $P={\left\{(Q_i,x_i)\right\}}_{i=1}^n;i=1,2,\dots ,n$ of Q, we have

$$\begin{array}{c}\hfill (L){\int }_{Q_i}{\mathtt{f}}_{2}d\mu \le (L){\int }_{Q_i}{\mathtt{f}}_{1}d\mu \le (L){\int }_{Q_i}{\mathtt{f}}_{3}d\mu .\end{array}$$

(4)

Therefore,

$$\begin{array}{c}\hfill (L){\int }_{Q_i}({\mathtt{f}}_2-ϵ)d\mu \le (L){\int }_{Q_i}{\mathtt{f}}_1(x_i)d\mu .\end{array}$$

Hence, $(L){\int }_{Q_i}{\mathtt{f}}_{2}d\mu -ϵ\mu (Q_i)\le {\mathtt{f}}_1(x_i)\mu (Q_i).$ In addition, ${\mathtt{f}}_1(x_i)\mu (Q_i)\le (L){\int }_{Q_i}{\mathtt{f}}_{3}d\mu +ϵ\mu (Q_i).$ Therefore, for $i=1,2,\dots ,n,$

$$\begin{array}{c}\hfill (L){\int }_{Q_i}{\mathtt{f}}_{2}d\mu -ϵ\mu (Q_i)\le {\mathtt{f}}_1(x_i)\mu (Q_i)\le (L){\int }_{Q_i}{\mathtt{f}}_{3}d\mu +ϵ\mu (Q_i).\end{array}$$

This gives

$$\begin{array}{c}\hfill (L){\int }_Q{\mathtt{f}}_{2}d\mu -ϵ\le \mathbb{S}({\mathtt{f}}_1,P)\le (L){\int }_Q{\mathtt{f}}_{3}d\mu +ϵ.\end{array}$$

(5)

From (4) and (5), $|\mathbb{S}({\mathtt{f}}_1,P)-(L){\int }_Q{\mathtt{f}}_{1}d\mu |<ϵ.$ Hence, ${\mathtt{f}}_1$ is a ${\mu }_{AP}$–Henstock–Kurzweil integrable on Q with respect to $\mu$ with $(L){\int }_Q{\mathtt{f}}_{1}d\mu =(AP){\int }_Q{\mathtt{f}}_{1}d\mu .$ □

4. AP–Henstock–Kurzweil Integral with Respect to Locally Finite Measures on Locally Compact Second Countable Hausdorff Spaces

We apply the fundamental findings from the theory of the ${\mu }_{AP}$–Henstock–Kurzweil integral to the case of bounded or locally finite measures on second countable Hausdorff spaces that are locally compact. A second countable locally compact Hausdorff space will now be represented by $\mathtt{X}$. Consider $\mathtt{X}={\displaystyle {\displaystyle \bigcup _{i\in \mathbb{N}}}}{\mathtt{X}}_i,$ where ${({\mathtt{X}}_i)}_{i\in \mathbb{N}}$ is an increasing sequence of relatively compact open subsets of $\mathtt{X}$ so that ${\mathtt{X}}_i\subseteq {\mathtt{X}}_{i+1}.$ Radon measures are always Borel regular by definition, but in general, the converse is not true. Indeed, for example, the counting measure n on $\mathtt{X},$ defined by letting $n(A)$ be the number of elements in $A,$ where $A\subset \mathtt{X}$ is Borel regular on any metric space $\mathtt{X}$, but it is a Radon measure only if every compact subset of $\mathtt{X}$ is finite, that is, $\mathtt{X}$ is discrete. A Borel measure $\mu$ on a locally compact Hausdorff space is regular if for all Borel subsets $\mathtt{B}$ of $\mathtt{X},$ we have

$$\begin{array}{c}\hfill \mu (\mathtt{B})=inf\{\mu (O):\mathtt{B}\subseteq O,O\mathrm{is}\mathrm{open}\}.\end{array}$$

That is,

$$\begin{array}{c}\hfill \mu (\mathtt{B})=sup\{\mu (\mathtt{K}):\mathtt{K}\subseteq \mathtt{B},\mathtt{K}\mathrm{is}\mathrm{compact}\}.\end{array}$$

In addition, any bounded Borel measure on a second countable locally compact Hausdorff space is regular. So, the Borel measure is equivalent to the Radon measure on a second countable locally compact Hausdorff space. Now, onwards, consider the Radon measure $\mu$ on a $\sigma$-algebra $\mathtt{M}$ of $\mathtt{X}$ endowed with a second countable locally compact Hausdorff topology $\mathfrak{T}\subseteq \mathtt{M}.$ We define the ${\mu }_{AP}$–Henstock–Kurzweil integrable relative to $\mu$ on Q as follows:

Definition 10. 

Let $\mathtt{f}:\overline{Q}\to \mathtt{R}$ be a function. $\mathtt{f}$ is said to be ${\mu }_{AP}$–Henstock–Kurzweil integrable relative to μ on Q if there exists a real number A such that for each $ϵ>0$, there exists a choice N-fine gauge Γ on Q such that

$$\left|\sum \mathtt{f}(x)\mu (Q_i)-A\right|<ϵ.$$

We denote the real number $A=(\overline{AP}){\int }_Q\mathtt{f}.$ A is called the ${\mu }_{AP}HK$ integral of $\mathtt{f}$ relative to μ on $Q.$

It is easy to see that the integral A is unique. An N-fine gauge on Q has to be outlined on $\overline{Q}$ not simply on Q since for each pair of sets for each $(Q_i,x_i)$ during a partition of $Q,$ the relevant point $x\in \overline{Q}$ while this is not true for $Q.$ The set of all functions that are ${\mu }_{AP}$–Henstock–Kurzweil integrable relative to $\mu$ on Q shall be denoted by ${\mu }_{rAP}HK(Q).$

We call a finite union of mutually disjoint $\mu$ sets as a $\mu$-elementary set (in short, an elementary set). Any subset of an elementary set is called a $\mu$-elementary subset (in short elementary subset). Say Q be an elementary set and $Q_0$ be an elementary subset of $Q.$ We call $Q_0$ a $\mu$-fundamental set (in short fundamental set) if $Q_0$ and $Q\backslash Q_0$ are $\mu$-elementary sets.

Remark 3. 

If $P={\left\{(Q_i,x_i)\right\}}_{i=1}^n$ is an N-fine partial division of Q which is not a division of $Q,$ then $Q\backslash {\displaystyle {\displaystyle \bigcup _{i=1}^n}}{Q}_i$ is necessarily an elementary set. This means for each $i=1,2,\dots ,n$ the set $Q\backslash Q_i$ is an elementary subset of Q and thus each $Q_i$ is a fundamental subset of $Q.$

Fundamental Characteristics

Within this subsection, we lay out a few basic characteristics of the $A{P}_{HK}$ integral. The main result here is a Saks–Henstock-type Lemma. In the sequel, Q is a $\mu$ set and $Q\in \mathtt{F}.$ Within this subsection, almost all x in $\overline{Q}$ means almost everywhere in $\overline{Q}.$ We consider a property is said to hold almost everywhere in $\overline{Q}$ if it holds everywhere except perhaps in a set of measure zero, that is that property holds for all $x\in \overline{Q}\backslash Y$ where $\mu (Y)=0.$

Proposition 5. 

Let Q be a μ set and $\mathtt{f}:\overline{Q}\to \mathtt{R}.$ If $\mathtt{f}(x)=0$ for almost all x in $\overline{Q},$ then $\mathtt{f}$ is ${\mu }_{rAP}$Henstock–Kurzweil integrable with the value 0 on $Q.$

Proof. 

Let $\mathtt{f}(x)=0$ for all $x\in \overline{Q}\backslash Y,$ where $\mu (Y)=0$ and Y is the union of ${\mathtt{X}}_i,i=1,2,\dots ,$ where ${\mathtt{X}}_i$ is a subset of Y such that $i-1\le |\mathtt{f}(x)|<i$ for $x\in {\mathtt{X}}_i.$ Each $\mu ({\mathtt{X}}_i)=0$ as $0\le \mu ({\mathtt{X}}_i)\le \mu (Y)=0.$ Given $ϵ>0$ and for each $i,$ using Proposition 2, we can choose an open set $U_i$ such that $\mu (U_i)<\frac{ϵ}{2^i\times i}$ and ${\mathtt{X}}_i\subseteq U_i.$ Let us define an N-fine gauge $\mathrm{\Gamma }$ on Q such that $\mathrm{\Gamma }(x)\subseteq U_i$ for $x\in {\mathtt{X}}_i,i=1,2,\dots .$ Then, for any N-fine partition $P={\left\{(Q_i,x_i)\right\}}_{i=1}^n$, we have

$$\begin{array}{cc}\hfill |\mathbb{S}(\mathtt{f},P)-0|& =\left|\sum \mathtt{f}(x_i)\mu (Q_i)-0\right|\hfill \\ & =\left|{\displaystyle \sum _{x_i\in {\overline{Q}}_i\backslash Y}}\mathtt{f}(x_i)\mu (Q_i)+{\displaystyle \sum _{x_i\in Y}}\mathtt{f}(x_i)\mu (Q_i)\right|\hfill \\ & \le {\displaystyle \sum _{i=1}^{\infty }}{\displaystyle \sum _{x_i\in {\mathtt{X}}_i}}\left|\mathtt{f}(x_i)\right|\mu (Q_i)\hfill \\ & <{\displaystyle \sum _{i=1}^{\infty }}\left(i\times \frac{ϵ}{2^i\times i}\right)\hfill \\ & =ϵ.\hfill \end{array}$$

□

Proposition 6. 

If ${\mathtt{f}}_1,{\mathtt{f}}_2\in {\mu }_{rAP}HK(Q)$, then for any scalars α and $\beta ,\alpha {\mathtt{f}}_1+\beta {\mathtt{f}}_2\in {\mu }_{rAP}HK(Q)$ and

$$(\overline{AP}){\int }_Q(\alpha {\mathtt{f}}_1+\beta {\mathtt{f}}_2)d\mu =\alpha (\overline{AP}){\int }_Q{\mathtt{f}}_{1}d\mu +\beta (\overline{AP}){\int }_Q{\mathtt{f}}_{2}d\mu .$$

Proposition 7. 

Let the functions ${\mathtt{f}}_1,{\mathtt{f}}_2\in {\mu }_{rAP}HK(Q)$ such that ${\mathtt{f}}_1(x)={\mathtt{f}}_2(x)$ almost everywhere in $\overline{Q},$ then ${\mathtt{f}}_1\in {\mu }_{rAP}HK(Q)$ if and only if ${\mathtt{f}}_2\in {\mu }_{rAP}HK(Q),$ and $(\overline{AP}){\int }_Q{\mathtt{f}}_{1}d\mu =(\overline{AP}){\int }_Q{\mathtt{f}}_{2}d\mu \mu$-a.e.

Proposition 8 

(Cauchy’s criterion). Let $\mathtt{f}$ be a real-valued function on $\overline{Q}.$ Then, $\mathtt{f}\in {\mu }_{rAP}HK(Q)$ if and only if for every $ϵ>0$, there exists an N-fine gauge Γ on Q such that for all N-fine partitions $P_1,P_2$ that are Γ-fine of $Q,$ it holds

$$\begin{array}{c}\hfill |\mathbb{S}(\mathtt{f},P_1)-\mathbb{S}(\mathtt{f},P_2)|<ϵ.\end{array}$$

Proof. 

The necessity follows from Definition 10. To prove the sufficiency, let ${\mathrm{\Gamma }}_n$ be an N-fine gauge on Q such that for each $n\in \mathbb{N}$ with each pair N-fine partitions $P_n^{{}^{\prime }}$ and $P_n^{{}^{″}}$ of $Q,$

$$\begin{array}{c}\hfill |\mathbb{S}(\mathtt{f},P_n^{\prime })-\mathbb{S}(\mathtt{f},P_n^{{}^{″}})|<ϵ.\end{array}$$

Let us consider an N-fine gauge ${\mathrm{\Gamma }}_{\Delta }=min\{{\mathrm{\Gamma }}_1(x),{\mathrm{\Gamma }}_2(x),\dots ,{\mathrm{\Gamma }}_n(x)\}$ on $Q.$ Then, there is a ${\mathrm{\Gamma }}_{\Delta }$-fine partition $P_n$ which is an N-fine of Q for each $n\in \mathbb{N}.$ If $m,n\in \mathbb{N},n<m:n\ge N.$ This implies: $P_n$ and $P_m$ are N-fine partitions of $Q.$ Hence,

$$\begin{array}{c}\hfill |\mathbb{S}(\mathtt{f},P_n)-\mathbb{S}(\mathtt{f},P_m)|<\frac{ϵ}{2}.\end{array}$$

This implies ${\left\{\mathbb{S}(\mathtt{f},P_n)\right\}}_{n=1}^{\infty }$ is a Cauchy sequence that converges to real number $A={\displaystyle \underset{n\to \infty }{lim}}\mathbb{S}(\mathtt{f},P_n).$ Then, $|\mathbb{S}(\mathtt{f},P_n)-A|<\frac{ϵ}{2}$ for each $n\ge N.$ On the condition that P is an N-fine partition on $Q,$ then $|\mathbb{S}(\mathtt{f},P)-A|<ϵ.$ Hence, $\mathtt{f}\in {\mu }_{rAP}HK(Q)$ and $A=(\overline{AP}){\int }_Q\mathtt{f}d\mu .$ □

Proposition 9. 

Let $\mathtt{f}$ be a real-valued function defined on $\overline{Q}.$ If $\mathtt{f}\in {\mu }_{rAP}HK(Q),$ then $\mathtt{f}\in {\mu }_{rAP}HK(Q_0)$ for every subset $Q_0$ of $Q.$

Proof. 

Let $Q_0\subset Q.$ Let $P_1={\left\{(Q_i,x_i)\right\}}_{i=1}^n$ and $P_2={\left\{(Q_j,x_j)\right\}}_{j=1}^m$ be N-fine partitions of $Q_0$ and let $P_3$ be an N-fine partition of $Q\backslash Q_0.$ It is very clear that $P_1\cap P_3=\varnothing ,P_2\cap P_3=\varnothing$ and $P_1\cup P_3,P_2\cup P_3$ are N-fine partitions of $Q.$ Then, by Cauchy’s criterion

$$\begin{array}{c}\hfill |\mathbb{S}(\mathtt{f},P_1\cup P_3)-\mathbb{S}(\mathtt{f},P_2\cup P_3)|<ϵ.\end{array}$$

Consequently, we obtain $|\mathbb{S}(\mathtt{f},P_1)-\mathbb{S}(\mathtt{f},P_2)|<ϵ$ so, $\mathtt{f}\in {\mu }_{rAP}HK(Q_0).$ □

Proposition 10. 

Let Q be a disjoint union of subsets $Q_1,Q_2,\dots ,Q_m.$ If $\mathtt{f}\in {\mu }_{rAP}HK(Q_i)$ for each $i=1,2,\dots ,m,$ then $\mathtt{f}\in {\mu }_{rAP}HK(Q)$ and $(\overline{AP}){\int }_Q\mathtt{f}d\mu ={\displaystyle {\displaystyle \sum _{i=1}^m}}(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu .$

Proof. 

The proof is very straightforward by using the Cousin’s lemma. □

Proposition 11. 

Let Y be a closed subset of $\overline{Q}.$ Then, ${\chi }_Y$ is ${\mu }_{rAP}$–Henstock–Kurzweil integrable on Q with the value $\mu (Y).$

Proof. 

Let $ϵ>0;$ then, from Proposition 2, there exists an open set U such that $Y\subseteq U$ and $\mu (U\backslash Y)<ϵ.$ Let $\mathrm{\Gamma }$ be an N-fine gauge on Q such that $\mathrm{\Gamma }(x)\subseteq U$ if $x\in Y$ and $\mathrm{\Gamma }(x)\subseteq \overline{Q}\backslash Y$ if $x\in \overline{Q}\backslash Y.$ Let $P={\left\{(Q_i,x_i)\right\}}_{i=1}^n$ be an N-fine partition of $Q.$ Then

$$\begin{array}{c}\hfill {\displaystyle \sum _{x\in Y}}{\chi }_Y(x)\mu (Q_i)={\displaystyle \sum _{x\in Y}}\mu (Q_i)\end{array}$$

and

$${\displaystyle \sum _{x\notin Y}}{\chi }_Y(x,\mu (Q_i))=0.$$

Let $Q_i$ be the union of subsets of Q such that $(Q_i,x_i)\in P$ and $x\in Y,$ then $Y\subseteq Q_i\subseteq U.$ So,

$$\begin{array}{cc}\hfill \left|\sum {\chi }_Y(x,\mu (Q_i))-\mu (Y)\right|& =\left|{\displaystyle \sum _{x\in Y}}\mu (Q_i)-\mu (Y)\right|\hfill \\ & =\mu (Q_i\backslash Y)\hfill \\ & \le ϵ.\hfill \end{array}$$

Hence the proof. □

Theorem 12 

(Saks–Henstock-type Lemma). Let $\mathtt{f}\in {\mu }_{rAP}HK(Q).$ For every $ϵ>0,$ there exists an N-fine gauge Γ on Q such that for any N-fine partition $P={\left\{(x_i,Q_i)\right\}}_{i=1}^n$ of Q, we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{i=1}^n}\left|\mathtt{f}(x_i)\mu (Q_i)-(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu \right|<ϵ.\end{array}$$

Proof. 

Let $ϵ>0$ be given and let $\mathrm{\Gamma }$ be an N-fine gauge on Q such that for any N-fine partition $P=\left\{(Q_i,x_i)\right\}$ on Q, we have

$$\begin{array}{c}\hfill |\sum \mathtt{f}(x_i)\mu (Q_i)-(\overline{AP}){\int }_Q\mathtt{f}d\mu |<ϵ.\end{array}$$

Now, by Proposition 9 and Remark 3, the integral $(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu$ exists for $i=1,2,\dots$ such that $\mathtt{f}(x_i)\mu (Q_i)-(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu \ge 0.$

By the ${\mu }_{rAP}HK$ integrability of $\mathtt{f}$ on each $Q_i,$ there exists an N-fine gauge ${\mathrm{\Gamma }}^{*}$ finer than $\mathrm{\Gamma },$ and N-fine (${\mathrm{\Gamma }}^{*}$-fine) partitions $P_1,P_2,\dots ,P_n$ on $Q_1,Q_2,\dots ,Q_n$, respectively, such that for $i=1,2,\dots$, we have

$$\begin{array}{c}\hfill |\sum \mathtt{f}(x_i)\mu (Q_i)-(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu |<\frac{ϵ}{n}.\end{array}$$

(6)

Now, using (6), it results

$$\begin{array}{cc}\hfill {\displaystyle \sum _{i=1}^n}\left|\mathtt{f}(x_i)\mu (Q_i)-(\overline{AP}){\int }_{Q_i}\mathtt{f}d\mu \right|& \le \left|\sum \mathtt{f}(x_i)\mu (Q_1)-(\overline{AP}){\int }_{Q_1}\mathtt{f}d\mu \right|+\dots \hfill \\ & +\left|\sum \mathtt{f}(x_i)\mu (Q_n)-(\overline{AP}){\int }_{Q_n}\mathtt{f}d\mu \right|\hfill \\ & <ϵ.\hfill \end{array}$$

□

5. Conclusions

The concept of an AP–Henstock–Kurzweil-type integral is given on a Cauchy metric measure space $\mathtt{X}$ with a Radon measure $\mu$ and a family of “sets” $\mathtt{F}$ that satisfy the Vitali covering theorem with respect to $\mu .$ The classical Henstock–Kurzweil integral on the real line is specifically enclosed by this integral. In this setting, Cauchy’s criterion of an AP–Henstock–Kurzweil integral is discussed. Finally, we extend this idea to second countable, locally compact Hausdorff spaces having bounded or locally finite measures. In this approach, the Saks–Henstock-type lemma is discussed. As a future research topic, we will investigate the validity of the converse of Proposition 4.