1 History and Aim

At the beginning of the 1900s, Denjoy and Perron developed descriptive processes for recovering a function from its derivative that solved known problems of classical Riemann and Lebesgue integrals. Many years later, an equivalent constructive Riemann-type integral process was developed by Henstock and Kurzweil. Both integration processes were generalized quite recently for many different spaces (see [1, 11] and [12]) solving the problem of recovering Fourier coefficients in Haar, Walsh and Vilenkin systems (see [9, 10, 14, 15] and [16]). Many properties of these non-absolute integrals were investigated, for example, the Hake property was studied with an abstract differential basis in a topological spaces, in terms of variational measure and in Riesz spaces (see [13, 17] and [2]).

To establish pointwise estimates for solutions of elliptic partial differential equations, in 1961 Calderon and Zygmund introduced the \(L^{r}\)-derivative (see [3]) and in 1968 L. Gordon described a Perron-type integral, the \(P_{r}\)-integral, that recovers a function from its \(L^{r}\)-derivative (see [4]). In 2004, Musial and Sagher extended the \(P_r\)-integral to the \(L^r\)-Henstock–Kurzweil integral, the \(HK_{r}\)-integral, that recovers also a function from its \(L^{r}\)-derivative (see [6]). Quite recently the integration by parts formula for the \(HK_{r}\)-integral was investigated by Musial and Tulone (see [7]) and the same authors described a norm on the space of \(HK_{r}\)-integrable functions and studied the dual and completion of this space (see [8]).

It is well known that the Henstock–Kurzweil integral is equivalent to the variational integral (see [5]). In this paper, we define the \(L^r\)-variational integral and we prove that it is equivalent to the \(HK_r\)-integral.

2 Introduction

We will assume that \(r\ge 1\) and we will consider the case of the closed interval \(\left[ a,b\right] \).

Definition 2.1

A function \(f:\left[ a,b\right] \rightarrow {\mathbb {R}}\) is \(L^{r}\)-variational integrable on \(\left[ a,b\right] \) if there exists a function \(F\in L^{r}\left[ a,b\right] \) with the following property: for each \(\varepsilon >0\) there exist a non-decreasing function \(\phi \) defined on \(\left[ a,b\right] \) and a gauge \(\delta \), i.e., a positive function, defined on \(\left[ a,b\right] \) such that \(\phi \left( b\right) -\phi \left( a\right) <\varepsilon \) and for any \(\delta \)-fine tagged interval \(\left( x,\left[ c,d\right] \right) \), where \(\left[ c,d\right] \subseteq \left[ a,b\right] ,\)

$$\begin{aligned} \left( \dfrac{1}{d-c}\int _{c}^{d}\left| F\left( y\right) -F\left( x\right) -f\left( x\right) \left( y-x\right) \right| ^{r}dy\right) ^{1/r}<\phi \left( d\right) -\phi \left( c\right) . \end{aligned}$$
(2.1)

We will use the following definition given in [6]

Definition 2.2

A function \(f:\left[ a,b\right] \rightarrow {\mathbb {R}} \) is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b\right] \) if there exists a function \(F\in L^{r}\left[ a,b\right] \) so that for any \( \varepsilon >0\) there exists a gauge \(\delta \) so that for any finite collection of nonoverlapping \(\delta \)-fine tagged intervals

$$\begin{aligned} {\mathcal {Q}}=\left\{ \left( x_{i},\left[ c_{i},d_{i}\right] \right) ,1\le i\le q\right\} , \end{aligned}$$

we have

$$\begin{aligned} \sum \limits _{i=1}^{q}\left( \dfrac{1}{d_{i}-c_{i}}\int _{c_{i}}^{d_{i}}\left| F\left( y\right) -F\left( x_{i}\right) -f\left( x_{i}\right) \left( y-x_{i}\right) \right| ^{r}dy\right) ^{1/r}<\varepsilon . \end{aligned}$$

By Theorem 5 in [6], the function F in the Definition 2.2 is unique up to an additive constant, so we can state that for each \(x\in \left( a,b\right] \)

$$\begin{aligned} F\left( x\right) =\left( HK_{r}\right) \int _{a}^{x}f. \end{aligned}$$

We need the following definition in a later theorem.

Definition 2.3

Let \(F\in L^{r}\left[ a,b\right] \). For \(x\in \left[ a,b\right] \) we say that F is \(L^{r}\)-continuous at x if

$$\begin{aligned} \lim _{h\rightarrow 0}\left( \dfrac{1}{2h}\int _{x-h}^{x+h}\left| F\left( y\right) -F\left( x\right) \right| ^{r}dy\right) ^{1/r}=0. \end{aligned}$$

If F is \(L^{r}\)-continuous for all \(x\in E\), we say that F is \(L^{r}\)-continuous on E.

The Henstock–Kurzweil integral primitive is continuous in the usual sense. In [6] is proved an equivalent result for \(L^r\)-Henstock–Kurzweil indefinite integral.

Theorem 2.4

The function F in the definition of the \(L^{r}\)-Henstock–Kurzweil is \(L^{r}\)-continuous on \(\left[ a,b\right] \).

Definition 2.5

Let \(\Phi \) be a function defined on the subintervals of \(\left[ a,b\right] \). The function \(\Phi \) is superadditive if

$$\begin{aligned} \Phi \left( \left[ u,v\right] \right) +\Phi \left( \left[ v,w\right] \right) \le \Phi \left( \left[ u,w\right] \right) , \end{aligned}$$

whenever \(a\le u<v<w\le b\). The function \(\Phi \) is continuous if for each \(c\in \left( a,b\right) \),

$$\begin{aligned} \lim _{x\rightarrow c^{-}}\Phi \left( \left[ x,c\right] \right) =0=\lim _{x\rightarrow c^{+}}\Phi \left( \left[ c,x\right] \right) \end{aligned}$$

and

$$\begin{aligned} \lim _{x\rightarrow b^{-}}\Phi \left( \left[ x,b\right] \right) =0=\lim _{x\rightarrow a^{+}}\Phi \left( \left[ a,x\right] \right) . \end{aligned}$$

Remark 2.6

Throughout this paper, if an interval function is said to be continuous, it is to be considered continuous in the sense of Definition 2.5.

Definition 2.7

Let \(\delta \) be a gauge and let

$$\begin{aligned} {\mathcal {P}}=\left\{ \left( x_{i},\left[ c_{i},d_{i}\right] \right) ,1\le i\le n\right\} \end{aligned}$$

be a \(\delta \)-fine partition of \(\left[ a,b\right] \). Let

$$\begin{aligned} W\left( {\mathcal {P}}\right) =\sum \limits _{i=1}^{q}\left( \dfrac{1}{ d_{i}-c_{i}}\int _{c_{i}}^{d_{i}}\left| F\left( y\right) -F\left( x_{i}\right) -f\left( x_{i}\right) \left( y-x_{i}\right) \right| ^{r}dy\right) ^{1/r}. \end{aligned}$$
(2.2)

The main tool we need to get the \(L^r\)-variational integral is the following definition of \(L^r\)-variation function.

Definition 2.8

For each subinterval \(\left[ c,d\right] \subseteq \left[ a,b\right] \) define

$$\begin{aligned} \Phi \left( \left[ c,d\right] \right) =\Phi \left( F,\delta ,\left[ c,d \right] \right) =\sup \left\{ W\left( {\mathcal {P}}\right) \right\} , \end{aligned}$$
(2.3)

where the supremum is taken over all \(\delta \)-fine partitions \({\mathcal {P}}\) of \(\left[ c,d\right] .\)

Theorem 2.9

The function \(\Phi \) is superadditive.

Proof

Let u, v and w be such that \(a\le u<v<w\le b\) and let \(\varepsilon >0\). If either \(\Phi \left( \left[ u,v\right] \right) =\infty \) or \(\Phi \left( \left[ v,w\right] \right) =\infty \) then surely \(\Phi \left( \left[ u,w \right] \right) =\infty \) and the assertion holds. Otherwise let \(\mathcal { P}_{1}\) be a partition of \(\left[ u,v\right] \) such that \(W\left( {\mathcal {P}} _{1}\right) >\Phi \left( \left[ u,v\right] \right) -\varepsilon \) and let \( {\mathcal {P}}_{2}\) be a partition of \(\left[ v,w\right] \) such that \(W\left( {\mathcal {P}}_{2}\right) >\Phi \left( \left[ v,w\right] \right) -\varepsilon .\) Let \({\mathcal {P}}={\mathcal {P}}_{1}\cup {\mathcal {P}}_{2}\), and clearly \( W\left( {\mathcal {P}}\right) =W\left( {\mathcal {P}}_{1}\right) +W\left( \mathcal { P}_{2}\right) .\) But \(W\left( {\mathcal {P}}\right) \le \Phi \left( \left[ u,w\right] \right) .\) Therefore,

$$\begin{aligned} \Phi \left( \left[ u,v\right] \right) +\Phi \left( \left[ v,w\right] \right) -2\varepsilon <W\left( {\mathcal {P}}_{1}\right) +W\left( {\mathcal {P}} _{2}\right) \le \Phi \left( \left[ u,w\right] \right) . \end{aligned}$$

\(\square \)

Now we can prove the following theorem that extends Theorem 11.9 in [5]

3 Main Results

Theorem 3.1

A function \(f: \left[ a,b\right] \rightarrow {\mathbb {R}} \) is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b\right] \) if and only if there exists a function \(F:\left[ a,b\right] \rightarrow {\mathbb {R}} \) with the following property: for each \(\varepsilon >0\) there exists a superadditive interval function \(\Phi \) defined on the subintervals of \( \left[ a,b\right] \) and a gauge \(\delta \) defined on \(\left[ a,b\right] \) such that \(\Phi \left( \left[ a,b\right] \right) <\varepsilon \) and for any \(\delta \)-fine tagged interval \(\left( x,\left[ c,d\right] \right) \), where \(\left[ c,d\right] \subseteq \left[ a,b\right] ,\)

$$\begin{aligned} \left( \dfrac{1}{d-c}\int _{c}^{d}\left| F\left( y\right) -F\left( x\right) -f\left( x\right) \left( y-x\right) \right| ^{r}dy\right) ^{1/r}<\Phi \left( \left[ c,d\right] \right) . \end{aligned}$$

Proof

Suppose there exists a function F with the property stated in the theorem. Let \(\varepsilon >0\) and choose \(\Phi \) and \(\delta \) according to the hypotheses. If \({\mathcal {P}}:=\left\{ \left( x_{i},\left[ c_{i},d_{i}\right] \right) ,1\le i\le n\right\} \) is a \(\delta \)-fine tagged partition of \( \left[ a,b\right] \), then

$$\begin{aligned}&\sum \limits _{i=1}^{n}\left( \dfrac{1}{d_{i}-c_{i}}\int _{c_{i}}^{d_{i}} \left| F\left( y\right) -F\left( x_{i}\right) -f\left( x_{i}\right) \left( y-x_{i}\right) \right| ^{r}dy\right) ^{1/r} \\&\quad \le \sum \limits _{i=1}^{n}\Phi \left( \left[ c_{i},d_{i}\right] \right) \le \Phi \left( \left[ a,b\right] \right) , <\varepsilon \end{aligned}$$

and so f is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b\right] .\)

Now suppose that f is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b \right] \) and let

$$\begin{aligned} F\left( x\right) =\left( HK_{r}\right) \int _{a}^{x}f, \end{aligned}$$

for each \(x\in \left( a,b\right] \). Let \(\varepsilon >0\). By hypothesis, there exists a gauge \(\delta \) on \(\left[ a,b\right] \) such that

$$\begin{aligned} \sum \limits _{i=1}^{n}\left( \dfrac{1}{d_{i}-c_{i}}\int _{c_{i}}^{d_{i}}\left| F\left( y\right) -F\left( x_{i}\right) -f\left( x_{i}\right) \left( y-x_{i}\right) \right| ^{r}dy\right) ^{1/r}<\varepsilon /2, \end{aligned}$$

whenever \({\mathcal {P}}\) is a \(\delta \)-fine tagged partition of \(\left[ a,b \right] \). Let

$$\begin{aligned} {\mathcal {P}}=\left\{ \left( x_{i},\left[ c_{i},d_{i}\right] \right) ,1\le i\le n\right\} \end{aligned}$$

and let \(W\left( {\mathcal {P}}\right) \) be defined as in (2.2) and let \(\Phi \) be defined on the subintervals of \(\left[ a,b\right] \) as in (2.3). By Theorem 2.9, \(\Phi \) is superadditive. Also,

$$\begin{aligned} \Phi \left( \left[ a,b\right] \right) \le \varepsilon /2<\varepsilon . \end{aligned}$$

Finally, by the definition of \(\Phi \), if \(\left( x,\left[ c,d\right] \right) \) is a \(\delta \)-fine tagged interval such that \(\left[ c,d\right] \subseteq \left[ a,b\right] ,\)

$$\begin{aligned} \left( \dfrac{1}{d-c}\int _{c}^{d}\left| F\left( y\right) -F\left( x\right) -f\left( x\right) \left( y-x\right) \right| ^{r}dy\right) ^{1/r}<\Phi \left( \left[ c,d\right] \right) . \end{aligned}$$

This completes the proof. \(\square \)

Theorem 3.2

A function \(f:\left[ a,b\right] \rightarrow \) is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b\right] \) if and only if f is \(L^{r}\)-variational integrable on \(\left[ a,b\right] .\)

Proof

Suppose first that f is \(L^{r}\)-variational integrable on \(\left[ a,b \right] .\) Let \(\varepsilon >0\) and let F, \(\delta \) and \(\phi \) satisfy the conditions in Definition 2.1. If \( {\mathcal {P}}=\left\{ \left( x_{i},\left[ c_{i},d_{i}\right] \right) ,1\le i\le n\right\} \) is a \(\delta \)-fine tagged partition of \(\left[ a,b\right] \), then

$$\begin{aligned}&\sum \limits _{i=1}^{n}\left( \dfrac{1}{d_{i}-c_{i}}\int _{c_{i}}^{d_{i}} \left| F\left( y\right) -F\left( x_{i}\right) -f\left( x_{i}\right) \left( y-x_{i}\right) \right| ^{r}dy\right) ^{1/r} \\&\quad \le \sum \limits _{i=1}^{n}\left( \phi \left( d_{i}\right) -\phi \left( c_{i}\right) \right) =\phi \left( b\right) -\phi \left( a\right) <\varepsilon \end{aligned}$$

and so f is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b\right] \) and

$$\begin{aligned} \left( HK_{r}\right) \int _{a}^{b}f=F\left( b\right) -F\left( a\right) . \end{aligned}$$

Now suppose that f is \(L^{r}\)-Henstock–Kurzweil integrable on \(\left[ a,b \right] \) and that for each \(x\in \left( a,b\right] \),

$$\begin{aligned} F\left( x\right) =\left( HK_{r}\right) \int _{a}^{x}f. \end{aligned}$$

Let \(\varepsilon >0\). By Theorem 3.1 there exists a superadditive interval function \(\Phi \) defined on \(\left[ a,b\right] \) such that \(\Phi \left( \left[ a,b\right] \right) <\varepsilon \) and

$$\begin{aligned} \left( \dfrac{1}{d-c}\int _{c}^{d}\left| F\left( y\right) -F\left( x\right) -f\left( x\right) \left( y-x\right) \right| ^{r}dy\right) ^{1/r}<\Phi \left( \left[ c,d\right] \right) , \end{aligned}$$

whenever \(\left( x,\left[ c,d\right] \right) \) is a \(\delta \)-fine tagged interval such that \(\left[ c,d\right] \subseteq \left[ a,b\right] \). Define \(\phi :\left[ a,b\right] \rightarrow {\mathbb {R}} \) by \(\phi \left( a\right) =0\) and \(\phi \left( x\right) =\Phi \left( \left[ a,x\right] \right) \) for all \(x\in \left( a,b\right] \). If \(a\le c<d\le b \), then

$$\begin{aligned} \phi \left( d\right) -\phi \left( c\right) =\Phi \left( \left[ a,d\right] \right) -\Phi \left( \left[ a,c\right] \right) \ge \Phi \left( \left[ c,d \right] \right) \ge 0 \end{aligned}$$

and so \(\phi \) is non-decreasing. In addition,

$$\begin{aligned} \phi \left( b\right) -\phi \left( a\right) =\Phi \left( \left[ a,b\right] \right) <\varepsilon . \end{aligned}$$

Suppose that \(\left( x,\left[ c,d\right] \right) \) is a \(\delta \)-fine tagged interval such that \(\left[ c,d\right] \subseteq \left[ a,b\right] \). Then,

$$\begin{aligned}&\left( \dfrac{1}{d-c}\int _{c}^{d}\left| F\left( y\right) -F\left( x\right) -f\left( x\right) \left( y-x\right) \right| ^{r}dy\right) ^{1/r} \\&\quad \le \Phi \left( \left[ c,d\right] \right) \le \phi \left( d\right) -\phi \left( c\right) . \end{aligned}$$

Hence, the function f is \(L^{r}\)-variational integrable on \(\left[ a,b \right] .\) This completes the proof. \(\square \)

Corollary 3.3

If f is \(L^{r}\)-variational integrable on \(\left[ a,b\right] \), then the function F which satisfies the conditions of Definition 2.1 is unique up to an additive constant.

We now prove the continuity of the interval function \(\Phi \).

Proposition 3.4

Let f be \(L^{r}\)-variational integrable on \(\left[ a,b\right] \) and let F be a function that satisfies (2.1). Let \(\delta \) be a gauge, \(\Phi =\Phi \left( \delta ,F\right) \) be as in (2.3), and assume that \(\Phi \left( \left[ a,b\right] \right) \) is finite. Then, \(\Phi \) is continuous.

Proof

We will prove that \(\hbox {lim}_{x\rightarrow c^{-}}\Phi \left( \left[ x,c\right] \right) =0\) for each \(c\in \left( a,b\right] \); the proof for right-handed limits is similar. Suppose by way of contradiction that \(\hbox {lim}_{x\rightarrow c^{-}}\Phi \left( \left[ x,c\right] \right) \) either fails to exist or exists and is not equal to zero. Since \(\Phi \) is nonnegative, there exists \(\eta >0\) such that \(\lim \sup _{x\rightarrow c^{-}}\Phi \left( \left[ x,c\right] \right) >\eta .\) Let us see that for every \(\xi \in \left[ a,c\right) ,\) \(\Phi \left( \left[ \xi ,c\right] \right) >\eta .\) Fix \(\xi \), there exists \(\xi<\zeta <c\) such that \(\Phi \left( \left[ \zeta ,c \right] \right) >\eta \). Since \(\Phi \) is superadditive, we have that

$$\begin{aligned} \Phi \left( \left[ \xi ,c\right] \right) \ge \Phi \left( \left[ \xi ,\zeta \right] \right) +\Phi \left( \left[ \zeta ,c\right] \right) \ge \Phi \left( \left[ \zeta ,c\right] \right) >\eta . \end{aligned}$$

Consequently, for each \(\xi \in \left[ a,c\right) ,\) there exists \(\mathcal {P }_{\xi }\), a \(\delta \)-fine tagged partition of \(\left[ \xi ,c\right] \) such that \(W\left( {\mathcal {P}}_{\xi }\right) >\eta \).

We now prove that we can make the following three assumptions about \( {\mathcal {P}}_{x}:\)

  1. 1.

    \({\mathcal {P}}_{x}\) contains at least two tagged intervals,

  2. 2.

    c is a tag of \({\mathcal {P}}_{x}\), and

  3. 3.

    the interval containing c is arbitrarily small.

Fix x and \(\varepsilon >0\). Choose \(y\in \left( \max \left( x,c-\varepsilon \right) ,c\right) \). By Cousin’s Lemma there exists \( {\mathcal {Q}}\), a \(\delta \)-fine tagged partition of \(\left[ x,y\right] \). Define \({\mathcal {P}}_{x}={\mathcal {Q}}\cup {\mathcal {P}}_{y}.\) We then have

$$\begin{aligned} W\left( {\mathcal {P}}_{x}\right) =W\left( {\mathcal {Q}}\right) +W\left( \mathcal { P}_{y}\right) \ge W\left( {\mathcal {P}}_{y}\right) >\eta . \end{aligned}$$

If c is the tag of its interval, then \({\mathcal {P}}_{x}\) has the desired properties.

Now suppose that c is not the tag of its interval. Let s and t be such that \(\left( t,\left[ s,c\right] \right) \) is the tagged interval which contains c. It is possible that \(s=t\) but we assume that \(t<c\).

It suffices to show that

$$\begin{aligned} \lim _{u\rightarrow c^{-}}W\left( \left\{ \left( t,\left[ s,u\right] \right) ,\left( c,\left[ u,c\right] \right) \right\} \right) =W\left( \left\{ \left( t,\left[ s,c\right] \right) \right\} \right) . \end{aligned}$$

Note that

$$\begin{aligned}&W\left( \left\{ \left( t,\left[ s,u\right] \right) ,\left( c,\left[ u,c \right] \right) \right\} \right) \\&\quad =\left( \dfrac{1}{u-s}\int _{s}^{u}\left| F\left( y\right) -F\left( t\right) -f\left( t\right) \left( y-t\right) \right| ^{r}dy\right) ^{1/r} \\&\qquad +\left( \dfrac{1}{c-u}\int _{u}^{c}\left| F\left( y\right) -F\left( c\right) -f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r}. \end{aligned}$$

Using Minkowski’s inequality, we have

$$\begin{aligned}&\left( \dfrac{1}{c-u}\int _{u}^{c}\left| F\left( y\right) -F\left( c\right) -f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r} \\&\quad =\left( \dfrac{1}{c-u}\right) ^{1/r}\left( \int _{u}^{c}\left| F\left( y\right) -F\left( c\right) -f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r} \\&\quad \le \left( \dfrac{1}{c-u}\right) ^{1/r}\left( \int _{u}^{c}\left| F\left( y\right) -F\left( c\right) \right| ^{r}dy\right) ^{1/r} \\&\qquad +\left( \dfrac{1}{c-u}\right) ^{1/r}\left( \int _{u}^{c}\left| f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r} \\&\quad \le \left( \dfrac{1}{c-u}\int _{u}^{c}\left| F\left( y\right) -F\left( c\right) \right| ^{r}dy\right) ^{1/r} \\&\qquad +\left| f\left( c\right) \right| \left( \dfrac{1}{c-u}\right) ^{1/r}\left( \int _{u}^{c}\left| \left( c-u\right) \right| ^{r}dy\right) ^{1/r} \\&\quad =\left( \dfrac{1}{c-u}\int _{u}^{c}\left| F\left( y\right) -F\left( c\right) \right| ^{r}dy\right) ^{1/r}+\left| f\left( c\right) \right| \left( c-u\right) . \end{aligned}$$

By Theorem 2.4 the function F is \(L^{r}\)-continuous at each point of \(\left[ a,b\right] \), and so we have that

$$\begin{aligned}&\lim _{u\rightarrow c^{-}}\left( \dfrac{1}{c-u}\int _{u}^{c}\left| F\left( y\right) -F\left( c\right) -f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r} \nonumber \\&\quad \le \lim _{u\rightarrow c^{-}}\left[ \left( \dfrac{1}{c-u} \int _{u}^{c}\left| F\left( y\right) -F\left( c\right) \right| ^{r}dy\right) ^{1/r}+\left| f\left( c\right) \right| \left( c-u\right) \right] =0. \end{aligned}$$
(3.1)

We also have that

$$\begin{aligned}&\lim _{u\rightarrow c^{-}}\left( \dfrac{1}{u-s}\int _{s}^{u}\left| F\left( y\right) -F\left( t\right) -f\left( t\right) \left( y-t\right) \right| ^{r}dy\right) ^{1/r} \\&\quad =\left( \dfrac{1}{c-s}\int _{s}^{c}\left| F\left( y\right) -F\left( t\right) -f\left( t\right) \left( y-t\right) \right| ^{r}dy\right) ^{1/r}. \end{aligned}$$

It follows that

$$\begin{aligned}&\lim _{u\rightarrow c^{-}}W\left( \left\{ \left( t,\left[ s,u\right] \right) ,\left( c,\left[ u,c\right] \right) \right\} \right) \\&\quad =\lim _{u\rightarrow c^{-}}W\left( \left\{ \left( t,\left[ s,u\right] \right) \right\} \right) =W\left( \left\{ \left( t,\left[ s,c\right] \right) \right\} \right) . \end{aligned}$$

We now prove the proposition. Set \(x_{1}=a\) and write

$$\begin{aligned} {\mathcal {P}}_{x_{1}}= & {} {\mathcal {Q}}_{1}\cup \left( c,\left[ x_{2},c\right] \right) \\ {\mathcal {P}}_{x_{2}}= & {} {\mathcal {Q}}_{2}\cup \left( c,\left[ x_{3},c\right] \right) \\&\vdots \\ {\mathcal {P}}_{x_{k}}= & {} {\mathcal {Q}}_{k}\cup \left( c,\left[ x_{k+1},c\right] \right) . \end{aligned}$$

By the result proved above, we may assume that for each k, \(c-x_{k}<1/k\) and, therefore, that \(x_{k}\rightarrow c.\)

For each n,  the collection

$$\begin{aligned} {\mathcal {P}}_{n}^{\prime }=\bigcup \limits _{k=1}^{n}{\mathcal {Q}}_{k} \end{aligned}$$

is a \(\delta \)-fine tagged partition of \(\left[ a,x_{n+1}\right] \). Hence,

$$\begin{aligned} W\left( {\mathcal {P}}_{n}^{\prime }\right) =\sum \limits _{k=1}^{n}W\left( {\mathcal {Q}}_{k}\right) \le \Phi \left( \left[ a,x_{n+1}\right] \right) \le \Phi \left( \left[ a,b\right] \right) <\infty . \end{aligned}$$

This shows that the series

$$\begin{aligned} \sum \limits _{k=1}^{\infty }W\left( {\mathcal {Q}}_{k}\right) \end{aligned}$$

converges and hence

$$\begin{aligned} \lim _{k\rightarrow \infty }W\left( {\mathcal {Q}}_{k}\right) =0. \end{aligned}$$

We then have for each k, 

$$\begin{aligned}&\eta <W\left( {\mathcal {P}}_{x_{k}}\right) \\&\quad =W\left( {\mathcal {Q}}_{k}\right) +\left( \frac{1}{c-x_{k+1}}\int _{x_{k+1}}^{c}\left| F\left( y\right) -F\left( c\right) -f\left( c\right) \left( y-c\right) \right| ^{r}dy\right) ^{1/r}. \end{aligned}$$

By (3.1), the term on the right goes to zero; therefore, the entire right side of the equality goes to zero. This contradiction completes the proof.

\(\square \)