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This article is cited in 1 scientific paper (total in 1 paper) On the P. Musiala, V. A. Skvortsovbc, P. Sworowskid, F. Tulonee a Chicago State University, Chicago, IL, USA b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia c Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia d Institute of Mathematics, Casimirus the Great University, Bydgoszcz, Poland e University of Palermo, Palermo, Italy
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    § 1. IntroductionThe well-known classical definition of the Lebesgue integral in terms of the absolute continuity of the indefinite integral is an example of a so-called full descriptive characterization of an integral. Definitions of this type are known also for a number of nonabsolute generalizations of the Lebesgue integral. The best known among them is Lusin’s definition of the Denjoy integral (see [1]). In general, we say that a descriptive characterization of a constructive integration process is obtained if a class of functions is found that coincides with the class of indefinite integrals of all functions integrable in the sense of this process, and if functions in this class are differentiable almost everywhere in a certain sense and the derivative is equal to the integrand almost everywhere. If the differentiability (in an appropriate sense, corresponding to the differentiability property of the indefinite integral) of functions in the class is an additional assumption imposed on the class, then we have only a partial descriptive characterization of the integral. On the other hand, if the corresponding differentiability property is a consequence of the description of the class and not an additional requirement, such a characterization is said to be a full descriptive characterization. For example, each continuous function in Lusin’s class $\mathrm{ACG}$ is differentiable almost everywhere in the sense of the approximate derivative, and this class gives a full descriptive characterization of the wide Denjoy integral ($\mathrm{D}$-integral; see [1]). At the same time, if we restrict this class by considering only continuous and almost everywhere differentiable (in the usual sense) $\mathrm{ACG}$-functions, then we obtain a partial descriptive characterization of the so-called Khintchine integral (see [2]). Descriptive characterizations of other non-absolute integrals were discussed in [3]–[9]. In this paper we obtain a full descriptive characterization of a Henstock–Kurzweil-type integral, the $\mathrm{HK}_r$-integral ( defined in 2004 by Musial and Sagher [10]), which serves to integrate a derivative defined in the space $L^r$, $1\leqslant r <\infty$. This $L^{r}$-derivative was introduced in [11] by Calderón and Zygmund to be used in some estimates for solutions of elliptic partial differential equations. The $\mathrm{HK}_r$-integral was defined as an extension of a Perron-type integral, the $\mathrm{P}_r$-integral, defined previously by Gordon [12], which also recovers a function from its $L^r$-derivative. The $\mathrm{HK}_r$-integral turned out to be strictly wider than the $\mathrm{P}_r$-integral (see [13] and [14]). It was also shown in [10] that the indefinite $\mathrm{HK}_r$-integral is $L^r$-differentiable almost everywhere and belongs to a Lusin-type class of $\mathrm{ACG}_r$-functions. Some other properties of this integral were investigated in [14]–[17]. Another class, also giving a descriptive characterization of the $\mathrm{HK}_r$-integral, was found in [18] in terms of the absolute continuity of the so-called $L^r$-variational measure generated by a function belonging to $L^r$. Both classes, the class of $\mathrm{ACG}_r$-functions and the class of functions generating absolutely continuous $L^r$-variational measure were known to coincide with the class of indefinite $\mathrm{HK}_r$-integrals, but only under the additional assumption that the functions are $L^r$-differentiable almost everywhere (see [18]). The problem of the $L^r$-differentiability almost everywhere of functions in these classes was left open. So, in fact, the descriptive characterization of the $\mathrm{HK}_r$-integral obtained in [10] and [18] was only partial. The main aim of the present paper is to prove that each $\mathrm{ACG}_r$-function is $L^r$-differentiable almost everywhere and thereby obtain a full descriptive characterization of the $\mathrm{HK}_r$-integral (see Theorem 3 in § 3). Next, in § 4, we construct an example of a function from a Lusin-type class $[\mathrm{ACG}]$ which, even under an additional assumption of continuity, fails to have the property of being $L^r$-differentiable almost everywhere for any ${r\geqslant1}$. Thereby we show that the class $[\mathrm{ACG}]$ is not contained in the class $\mathrm{ACG}_r$ for any $r$. As a byproduct of this example, we obtain a new proof, based on the $L^{r}$-derivative, of a result obtained previously in [19] and stating that the $\mathrm{HK}_r$-integral does not cover the wide Denjoy integral. § 2. PreliminariesWe recall here the main definitions and facts related to the notion of the $\mathrm{HK}_r$-integral (see [10] and [12]). We work in a fixed segment (that is, closed interval) $[a,b]$. We let $I$ denote a nondegenerate subsegment of $[a,b]$. We assume in all definitions that follow that the function $F$ belongs to the class $L^r=L^r[a,b]$, $1\leqslant r<\infty$. First let us define $L^r$-derivates and the $L^r$-derivative. Definition 1. The upper-right $L^r$-derivate of $F$ at $x\in[a,b)$, denoted by $D_r^+F(x)$, is defined as the infimum of all numbers $\alpha$ such that We need the following property of $L^r$-derivates (see [12], the corollary to Theorem 4). Proposition 1. If the two upper (or two lower) $L^r$-derivates of a function $F$ are less than $+\infty$ (greater than $-\infty$, respectively) on a set $E$, then almost everywhere on $E$ the $L^{r}$-derivative of $F$ exists and is finite. We recall also the following definitions; see [1] and [7] for a full treatment. Definition 2. Let $F\colon[a,b]\to\mathbb R$, and let $c\in(a,b)$. The function $F$ is said to be approximately continuous at $c$ if there exists a measurable set $E\subset[a,b]$, $c\in E$, such that $c$ is a point of density of $E$ and $F|_E$ is continuous at $c$. Definition 3. Let $F\colon[a,b]\to\mathbb R$, and let $c\in(a,b)$. The function $F$ is said to be approximately differentiable at $c$ if there exists a measurable set $E\subset[a,b]$ such that $c$ is a point of density of $E$ and the limit is finite. The value of this limit is the approximate derivative of function $F$ at $c$ and is denoted by $F'_{\mathrm{ap}}(c)$. Note that the value of $F'_{\mathrm{ap}}(c)$ does not depend on the choice of $E$. Definitions 2 and 3 are formulated accordingly in the cases $x=a$ and $x=b$. In order to define the $\mathrm{HK}_r$-integral we need some auxiliary notions. A tagged interval is a pair $(I,x)$ where a point $x\in I$ is a tag. We say that two tagged intervals $(I',x')$ and $(I'',x'')$ are nonoverlapping if the intervals $I'$ and $I''$ are nonoverlapping, that is, they have no interior points in common. A partition is an arbitrary finite collection $\pi$ of pairwise nonoverlapping tagged intervals. A gauge is a strictly positive function $\delta$ on $[a,b]$ (or on a subset of $[a,b]$). We say that a tagged interval $(I, x)$ is $\delta$-fine if $I\subset(x-\delta(x),x+\delta(x))$. A partition is $\delta$-fine if all of its elements are $\delta$-fine. A partition $\pi$ is tagged in a set $E\subset [a,b]$ if $x \in E$ for all tagged intervals $(I, x)$ of $\pi$. The Lebesgue measure on $[a,b]$ will be denoted by $\mu$. We recall the definition of the $L^{r}$-Henstock–Kurzweil integral given in [10]. Definition 4. A function $f\colon[ a,b] \to \mathbb{R}$ is $L^{r}$-Henstock–Kurzweil integrable ($\mathrm{HK}_r$-integrable) if there exists a function $F\in L^{r}$ such that for any $\varepsilon >0$ there exists a gauge $\delta$ such that for any $\delta$-fine partition $\{([c_{i},d_{i}], x_i)\}_i$ we have Moreover, as shown in [10], the function $F$ is unique up to an additive constant, so that setting $F(a)=0$ one can consider the indefinite $\mathrm{HK}_r$-integral
$$
\begin{equation*}
F( x) =(\mathrm{HK}_r) \int_{a}^{x}f, \qquad x\in [a,b].
\end{equation*}
\notag
$$
Definition 5 ([10]). Let $E\subset[a,b]$ and $F\in L^r$. We say that $F$ is an $\mathrm{AC}_r(E)$-function, or $F\in\mathrm{AC}_r(E)$, if for each $\varepsilon>0$ there exist $\eta>0$ and a gauge $\delta$ defined on $E$ such that for any $\delta $-fine partition $\{([c_i,d_i], x_i)\}_i$ tagged in $E$ and satisfying we have We say that $F$ is an $\mathrm{ACG}_r$-function if $[a,b]=\bigcup_{n=1}^\infty E_n$ where $F\in\mathrm{AC}_r(E_n)$ for all $n$. Definition 6. Let $E\subset [a,b]$. We say that $F$ is an $\mathrm{AC}(E)$-function, or $F\in\mathrm{AC}(E)$, if for each $\varepsilon>0$ there exists $\eta>0$ such that for any finite collection of nonoverlapping segments $\{[c_i,d_i]\}_i$ with endpoints in $E$ such that we have We say that $F$ is an $\mathrm{ACG}$-function if $[a,b]=\bigcup_{n=1}^\infty E_n$ and $F\in\mathrm{AC}(E_n)$ for all $n$. If, moreover, all the $E_n$ can be chosen closed, we say $F$ is an $[\mathrm{ACG}]$-function. Definition 7. Let $E\subset [a,b]$. We say that $F$ is a $\mathrm{VB}(E)$-function, or $F\in\mathrm{VB}(E)$, if there is a finite upper bound for the sums $\sum_i|F(d_i)-F(c_i)|$ over arbitrary finite collections of nonoverlapping segments $\{[c_i,d_i]\}_i$ with endpoints in $E$. We say that $F$ is a $\mathrm{VBG}$-function if $[a,b]=\bigcup_{n=1}^\infty E_n$ and $F\in\mathrm{VB}(E_n)$ for all $n$. Definition 8. A function $f\colon[a,b]\to\mathbb R$ is said to be $\mathrm{D}$-integrable (integrable in the wide sense of Denjoy) if there exists a continuous $\mathrm{ACG}$-function $F\colon[a,b]\to\mathbb R$ such that $F'_{\mathrm{ap}}(x)=f(x)$ at almost all $x\in[a,b]$. We then define $\int_a^bf=F(b)-F(a)$. Definition 9. A function $f\colon[a,b]\to\mathbb R$ is said to be Kubota integrable if there exists an approximately continuous $[\mathrm{ACG}]$-function $F\colon[a,b]\to\mathbb R$ such that $F'_{\mathrm{ap}}(x)=f(x)$ at almost all $x\in[a,b]$. We then define $\int_a^bf=F(b)-F(a)$. We will use the following property of $\mathrm{ACG}_r$-functions. Proposition 2 (see [10], Corollary 1, and [19], Corollary 13). Each $\mathrm{ACG}_r$-function is also an $\mathrm{ACG}$-function. It is known (see [1], Ch. VII) that each $\mathrm{ACG}$-function is a $\mathrm{VBG}$-function. Since $F\in L^r$, $F$ is measurable, and so from the above proposition it follows that any $\mathrm{ACG}_r$-function is a measurable $\mathrm{VBG}$-function. Hence by the Denjoy–Khintchine theorem (see [1], Ch. VII, Theorem 4.3) we obtain the following result. Theorem 1. Each $\mathrm{ACG}_r$-function is approximately differentiable almost everywhere on $[a,b]$. § 3. $L^r$-differentiability of $\mathrm{ACG}_r$-functionsWe now prove the main theorem in this paper. Proof. According to Theorem 1, $F'_{\mathrm{ap}}$ exists and is finite almost everywhere on $[a, b]$. We will show that By Lusin’s theorem on the $C$-property it is sufficient to prove (3.1) on a closed set $E$ of positive measure such that $F$, as restricted to $E$, is continuous. We can suppose that $F$ is approximately differentiable at each point in $E$. Note that the approximate derivative and all $L^r$-derivates are measurable (see [1] and [12]). To simplify the computation we consider first the case where $F'_{\mathrm{ap}}$ is nonnegative on $E$. We start by proving that in this case
$$
\begin{equation}
D_r^{+}F(x)=F_{\mathrm{ap}}'(x) \quad\text{a.e. on }E.
\end{equation}
\tag{3.2}
$$
If not, then having in mind the relation between $L^r$-derivates and approximate derivates (see [12], Theorem 2 and 3) we obtain on a set $E'\subset E$ of positive measure. We can suppose that $E'$ is closed. By Definition 5 we can find a set $E''\subset E'$ of positive outer measure for which ${F\in\mathrm{AC}_r(E'')}$. By Definition 5 for $\varepsilon=1$ there exist a gauge $\delta$ on $E''$ and a positive number $\eta$ such that inequality (2.4) holds for each $\delta$-fine partition $\{([c_i,d_i],x_i)\}_i$ tagged in $E''$ and satisfying (2.3). Now fixing $\delta$ corresponding to the chosen $\varepsilon$ we can find a set $T \subset E''$ of positive outer measure on which $\delta(x)\geqslant \gamma$ for some positive constant $\gamma$. Let $\overline{T}$ be the closure of $T$, and let $W$ be the set of two-sided limit points of $\overline{T}$. Then we have $\overline{T}=W\cup C$, where $C$ is countable. As $E'$ is closed, we have $W\subset\overline{T}\subset E'\subset E$ and $\mu(W)>0$.
We show now that inequality (2.4) for $\varepsilon=2$, that is, the inequality
$$
\begin{equation}
\sum_i\biggl(\frac1{d_i-c_i}\int_{c_i}^{d_i}|F(y)-F(x_i)|^r\,dy\biggr)^{1/r}<2
\end{equation}
\tag{3.4}
$$
holds for any $\gamma$-fine partition $\{([c_i,d_i],x_i)\}_i$ tagged in $W$ and satisfying (2.3) for the same $\eta$ as found for $\varepsilon=1$ and $\delta$-fine partitions tagged in $E''$. Indeed, consider such a $\gamma$-fine partition. By the definition of $W$ and by the continuity of $F$ with respect to $E$, for each $i$ we can choose a point $x_i'\in T$ sufficiently close to $x_i$ and inside $[c_i,d_i]$, so that $\{([c_i,d_i],x_i')\}$ is also a $\gamma$-fine partition, for which (2.4) holds for $\varepsilon=1$, and $\sum_i|F(x_i) - F(x'_i)|<1$. Then
$$
\begin{equation*}
\begin{aligned} \, &\sum_i\biggl( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|F(y)-F(x_i)|^r\, dy\biggr)^{1/r} \\ &\qquad\leqslant\sum_i\biggl( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|F(y)-F(x_i')|^r\, dy\biggr)^{1/r} \\ &\qquad\qquad+\sum_i\biggl( \frac{1}{d_i-c_i}\int_{c_i}^{d_i}|F(x_i')-F(x_i)|^r\, dy\biggr)^{1/r}<1+1=2. \end{aligned}
\end{equation*}
\notag
$$
Hence (3.4) must be true for any $\gamma$-fine partition tagged in $W$ and satisfying (2.3). We will now construct however a $\gamma$-fine partition for which (3.4) fails to be true. This leads to a contradiction, which proves (3.2).
Let $P\subset W$ be the set of density points of $W$. We have $\mu(P)=\mu(W)$. Let $x$ be a point of $P$. There exists a set $S_x\subset[a,b]$ having $x$ as a density point, such that
$$
\begin{equation}
\lim_{\substack{t\to 0\\x+t\in S_x}}\frac{F(x+t) - F(x)}{t}= F_{\mathrm{ap}}'(x)\geqslant0.
\end{equation}
\tag{3.5}
$$
Note that $x$ is also a density point for the set $W\cap S_x$. Moreover, we can find a closed set $A_x\subset W\cap S_x$ for which $x$ is still a density point. Let the open set $B_x$ be the complement of $A_x$ in $[a,b]$. Choose a number $l$ so that $D^+_r(x)>l>F_{\mathrm{ap}}'(x)$. Then by Definition 1 there exist a positive constant $c_x$ and a sequence $h_k\to 0^+$ such that for each $k$
$$
\begin{equation}
\biggl(\frac1{h_k}\int_{0}^{h_k}[F(x+t)-F(x)-lt]_+^r\, dt\biggr)^{1/r}>c_x h_k.
\end{equation}
\tag{3.6}
$$
We now put some restrictions on the size of $h_k$. Let us cover $P$ by an open set $G\subset(a,b)$ such that $\mu(G)<2\mu(P)$, and assume that all intervals $[x,x+h_k]$ are subsets of $G$. As $A_x\cap[x,x+h]\subset\{x+t\colon F(x+t)-F(x)<lt\}$ for sufficiently small $h$, we can also assume that for each $h_k$
$$
\begin{equation}
\begin{aligned} \, &\int_0^{h_k}[F(x+t)-F(x)-lt]_+^r\, dt \nonumber \\ &\qquad=\int_{B_x\cap[x, x+h_k]}[ F(u)-F(x)-l(u-x)]_+^r\, du. \end{aligned}
\end{equation}
\tag{3.7}
$$
Note that this equality implies, in particular, that in our further considerations we can assume that for all $k$, Indeed, otherwise contradicting the claim that $D^+_rF(x)>l$.
Fix some natural number $n>\max{\{2\mu(P)/\eta,2/\mu(P)\}}$. As $x$ is a point of dispersion for the set $B_x$, we can find $h_x\leqslant\gamma$ such that
$$
\begin{equation}
\frac{\mu(B_x\cap[x, x+h])}{h}< \min\biggl\{\frac1n,\frac{c_x^r}{n^r}\biggr\}
\end{equation}
\tag{3.9}
$$
for $0<h\leqslant h_x$. So we can assume that $h_k<h_x\leqslant\gamma$ for all $k$. Having in mind (3.6)–(3.9), we obtain
$$
\begin{equation}
\begin{aligned} \, &\biggl(\frac1{\mu(B_x\cap[x,x+h_k])}\int_{B_x\cap[x, x+h_k]}[F(u)-F( x) -l(u-x)]_+^r\, du\biggr)^{1/r} \nonumber \\ &\qquad =\biggl(\frac{h_k}{\mu(B_x\cap[x,x+h_k])}\biggr)^{1/r} \biggl(\frac1{h_k}\int_0^{h_k}[F(x+t)-F(x)-lt]_+^r\, dt\biggr)^{1/r} \nonumber \\ &\qquad \geqslant\biggl(\frac{n^r}{c^{r}_x}\biggr)^{1/r}c_xh_k=nh_k. \end{aligned}
\end{equation}
\tag{3.10}
$$
As $B_x$ is an open set, we have a representation Note that $a_j\in A_x\subset W$ and $b_j-a_j<h_x\leqslant\gamma$ for each $j$. So the $([a_j,b_j],a_j)$ are $\gamma$-fine intervals tagged in $W$. Using the following obvious inequality for positive numbers:
$$
\begin{equation*}
\biggl(\frac{\sum_ka_k}{\sum_kb_k}\biggr)^{1/r} \leqslant\biggl(\sum_k\frac{a_k}{b_k}\biggr)^{1/r}\leqslant\sum_k\biggl(\frac{a_k}{b_k}\biggr)^{1/r}
\end{equation*}
\notag
$$
(this is true for infinite sums, provided the series are convergent), we obtain
$$
\begin{equation}
\begin{aligned} \, &\biggl(\frac1{\mu(B_x\cap[x,x+h_k])}\int_{B_x\cap[x,x+h_k]}[F(u)-F(x)-l(u-x)]_+^r\, du\biggr)^{1/r} \nonumber \\ &\qquad\leqslant\biggl(\sum_j\frac1{b_j-a_j}\int_{a_j}^{b_j}[F(u)-F(x)-l(u-x)]_+^r\, du\biggr)^{1/r} \nonumber \\ &\qquad\leqslant\biggl(\sum_j\frac1{b_j-a_j}\int_{a_j}^{b_j}|F(u) -F(a_j)|^r\, du\biggr)^{1/r} \nonumber \\ &\qquad\leqslant\sum_j\biggl(\frac1{b_j-a_j}\int_{a_j}^{b_j}|F(u)-F(a_j)|^r\, du\biggr)^{1/r} \end{aligned}
\end{equation}
\tag{3.11}
$$
(we have taken into account that $F(a_j)- F(x)<l(a_j-x)<l(u-x)$ for $u\in(a_j,b_j)$). Combining (3.11) with (3.10) we obtain
$$
\begin{equation}
\sum_j\biggl(\frac 1{b_j-a_j}\int_{a_j}^{b_j}|F(u) -F(a_j)|^{r}\, du\biggr)^{1/r}\geqslant nh_k.
\end{equation}
\tag{3.12}
$$
The family of intervals $\{[x,x+h_k]\}$ for all $x\in P$ and all $k$ forms a Vitali cover of $P$. Applying the Vitali covering theorem we choose a countable sequence of nonoverlapping segments $[x_j,x_j+h_{k_j}]$ contained in $G$ and covering $P$ up to a set of measure zero. Hence $\mu(P)\leqslant\sum_jh_{k_j}\leqslant\mu(G)<2\mu(P)$. We can apply (3.12) to each of these segments and to the corresponding set $B_{x_j}\cap[x_j, x_j+ h_{k_j}]$. Collecting all intervals constituting $B_{x_j}\cap[x_j, x_j+ h_{k_j}]$ for all $j$ we obtain a family $\{(I_i,z_i)\}_i$ of nonoverlapping $\gamma$-fine intervals tagged in $P$, and therefore in $W$, the total length of which can be estimated, due to (3.9), by
$$
\begin{equation*}
\sum_j\mu(B_{x_j}\cap[x_j, x_j+ h_{k_j}]) < \sum_j\frac{ h_{k_j}}n \leqslant\frac{ 2\mu(P)}n<\eta.
\end{equation*}
\notag
$$
So inequality (3.4) must hold for any finite subfamily of $\{(I_i,z_i)\}_i$. At the same time, by (3.12) and the choice of $n$ we have
$$
\begin{equation*}
\sum_{i} \biggl(\frac 1{|I_{i}|}\int_{I_{i}}|F(u)-F(z_{i})|^r\,du\biggr)^{1/r}>n\mu(P)>2,
\end{equation*}
\notag
$$
which must also hold for a certain finite subfamily of $\{(I_i,z_i)\}_i$ (that is, a $\gamma$-fine partition tagged in $W$). This contradiction proves that our assumption (3.3) fails and (3.2) is true in the case of nonnegative $F'_{\mathrm{ap}}$.
In the same way, using the definition of the upper-left $L^r$-derivate we obtain a contradiction to the assumption that the inequality holds on a set of positive measure. In this case, instead of (3.7) the equality
$$
\begin{equation}
\begin{aligned} \, &\int_{0}^{h_k}[F(x-t)-F(x)+lt] _{-}^{r}\, dt \nonumber \\ &\qquad=\int_{B_x\cup[x-h_k,x]}[F(u)-F(x)-l(u-x)]_-^r\, du \end{aligned}
\end{equation}
\tag{3.14}
$$
is used for appropriate $h_k$. This time the intervals $\{(a_i,b_i)\}_i$ constituting $B_x\cap[{x-h_k},x]$ give rise to a $\gamma$-fine partition $\{(I_i,z_i)\}_i$ tagged in $W$. Eventually, we obtain
$$
\begin{equation}
D_r^{-}F(x)=F_{\mathrm{ap}}'(x) \quad\text{a.e. on }E
\end{equation}
\tag{3.15}
$$
(again, in the case when $F'_{\mathrm{ap}}(x)\geqslant0$). Having (3.2) and (3.15), we apply Proposition 1 and so prove the theorem in the case of nonnegative $F'_{\mathrm{ap}}$.
If $F'_{\mathrm{ap}}$ is negative on a set of positive measure on which (3.1) is false, we apply the argument above to the function $-F$. Theorem 2 is proved. As we have already noted, the above theorem shows that the class of $\mathrm{ACG}_r$-functions coincides with the class of indefinite integrals of all $\mathrm{HK}_r$-integrable functions, and so provides a full descriptive characterization of the $\mathrm{HK}_r$-integral. Theorem 3. A function $f$ is $\mathrm{HK}_r$-integrable on $[a,b]$ if and only if there exists an $\mathrm{ACG}_r$-function $F$ such that $F_r'=f$ almost everywhere; moreover, the function $F(x)-F(a)$ is the indefinite $\mathrm{HK}_r$-integral of $f$. § 4. The $\mathrm{ACG}$ class and $L^r$-differentiabilityThe following statement was proved in [19], Corollary 13. Using this proposition and equality (3.1) for an indefinite $\mathrm{HK}_r$-integral in combination with its approximate continuity (see [10]), we obtain the following result. Theorem 4. Every $\mathrm{HK}_r$-integrable function on $[a,b]$ is Kubota integrable (see Definition 9) on $[a,b]$, and the values of these integrals coincide. Once again using Proposition 3 and Theorem 3 we obtain a slight improvement of Proposition 2. We now show that an $\mathrm{ACG}$-function, even a continuous one, can fail to be $L^r$-differentiable almost everywhere. This, in particular, will imply that the inclusion stated in Corollary 1 is proper. Theorem 5. There exists an $\mathrm{ACG}$-function which is not $L^r$-differentiable on a set of positive measure for any $r$. Proof. Let $r_{0,1}=[0,1]$. We remove from $r_{0,1}$ the open interval $u_{0,1}$ centred at $1/2$ and of length $u_0=1/6$, leaving two segments $r_{1,1}$ and $r_{1,2}$, of length $r_1=5/12$ each. We continue this process in such a way that at step $n$ we are left with $2^n$ segments $\{r_{n,k}\}_{1\leqslant k\leqslant2^n}$, having length $r_n:=(n+4)/(2^{n+1}(n+2))$ each, and so In order to complete the recursive construction, from each interval $r_{n,k}$ we delete the concentric open interval $u_{n,k}$ of length $u_n:=1/(2^{n}(n+2)(n+3))$. The set is a symmetric, perfect Cantor-like set of measure $1/2$. Let $v_{n,k}$ be the open interval concentric with $u_{n,k}$ and having length $v_n:=u_n/2$. We define a function $F$ taking the value $0$ on $P$ and $(-1)^n/n$ on $v_{n,k}$. Then we extend $F$ linearly to the subintervals of $u_{n,k}$ adjacent to $v_{n,k}$.
The function $F$ is obviously continuous, and it is in $\mathrm{AC}(P)$ and $\mathrm{AC}(u_{n,k})$ for each $n$ and $k$. So it is an $\mathrm{ACG}$-function on $[0,1]$. We show now that $F$ fails to be $L^r$-differentiable at any point of $P$ which is not right-hand isolated in $P$. Let $x$ be such a point. A suitably large $n\in\mathbb N$ can be found such that for some $h>0$ and $k$, $x+h$ is the midpoint of $v_{n,k}$. Let $l>0$ be such that $x+l$ is the left-hand endpoint of such $v_{n,k}$ and consider any $\alpha\in\mathbb R$. Then we can estimate
$$
\begin{equation*}
\begin{aligned} \, &\frac1h\biggl(\frac1h\int_0^h|F(x+t)-F(x)-\alpha t|^r\,dt\biggr)^{1/r} \\ &\qquad >\frac1h\biggl(\frac1h\int_l^h|F(x+t)-\alpha t|^r\,dt\biggr)^{1/r}\geqslant\frac1h\biggl(\frac1h\int_l^h|F(x+t)|^r\,dt\biggr)^{1/r}, \end{aligned}
\end{equation*}
\notag
$$
where $n$ is odd when $\alpha\geqslant0$ and $n$ is even when $\alpha\leqslant0$. Continuing this estimate we obtain
$$
\begin{equation*}
\begin{aligned} \, &\frac1h\biggl(\frac1h\int_0^h|F(x+t)-F(x)-\alpha t|^r\,dt\biggr)^{1/r} \\ &\qquad \geqslant\frac2{r_n}\biggl(\frac2{r_n}\,\frac{v_n}{2n^r}\biggr)^{1/r} =\frac{2^{1-1/r}}{nr_n}\biggl(\frac{u_n}{r_n}\biggr)^{1/r} \\ &\qquad =2^{n+2}\frac{n+2}{n(n+4)^{1+1/r}(n+3)^{1/r}}, \end{aligned}
\end{equation*}
\notag
$$
which is unbounded in $n$. This proves that the function $F$ fails to be $L^r$-differentiable on a set of measure $1/2$.
The theorem is proved. As we have already mentioned, this example gives another proof of the following result obtained in [19], Theorem 14. Corollary 2. There exists a function that is $\mathrm{D}$-integrable but which is $\mathrm{HK}_r$-integrable for no $r$. Indeed, suppose that $f=F'_\mathrm{ap}$ is $\mathrm{HK}_r$-integrable ($F$ comes from the proof of Theorem 5). Then the indefinite $\mathrm{HK}_r$-integral $\displaystyle G=\int f$ is an $[\mathrm{ACG}]$-function (see Proposition 3) such that $G'_r(x)=f(x)$ almost everywhere in $[0,1]$ (Theorem 3). Since $G'_\mathrm{ap}(x)=f(x)$ almost everywhere in $[0,1]$, by a monotonicity theorem for the class of approximately continuous $[\mathrm{ACG}]$-functions (see [20], Theorem 1) $F-G$ is a constant, so that $F$, like $G$, is $L^r$-differentiable almost everywhere, producing a contradiction. 
Citation in format AMSBIB
\Bibitem{MusSkvSwo25} |
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