A Stieltjes type extension of the \(L^{r}\)-Perron integral (Q1704437)

In this paper, the authors give a Stieltjes-type extension to the \(L^{r}\)-Perron integral of \textit{L. Gordon} [Stud. Math. 28, 295--316 (1967; Zbl 0152.24605)], as follows. Definition 1. Suppose that \(f, \Phi:[a, b]\to \mathbb{R}\), where \(\Phi\) is a monotone increasing Lipschitz function. If \(\inf \Psi(b)\) taken over all \(L^{r;\Phi}\)-ex-major functions of \(f\), equals \(\sup \lambda(b)\) taken over all \(L^{r;\Phi}\)-ex-minor functions of \(f\), then the common value, denoted by \((P_{r, \Phi})\int f\), is called the \(P_{r, \Phi}\)-integral of \(f\) on \([a, b]\). Here, \(\Psi\in L^{r}[a, b]\) finite valued, is called a \(L^{r;\Phi}\)-ex-major function of \(f\), if \(\Psi(a)=0\), \(\Psi\) is \(L^{r}\)-continuous on \([a, b]\) and except for at most a denumerable subset of \([a, b]\), we have \(-\infty\not={\underline{D}}_{r}\Psi(x; \Phi)\geq f(x)\). If \(-\lambda\) is a \(L^{r;\Phi}\)-ex-major function of \(f\), then \(\lambda\) is called a \(L^{r;\Phi}\)-ex-minor function of \(f\). Also, \({\underline{D}}_{r}\Psi(x; \Phi)\) defines the lower (two-sided) \(L^{r;\Phi}\)-derivate as \({\underline{D}}_{r}\Psi(x; \Phi)=\min\{D_{+, r}\Psi(x;\Phi), D_{-, r}\Psi(x;\Phi)\}\), where \(D_{+, r}\Psi(x;\Phi)\) and \(D_{-, r}\Psi(x;\Phi)\) denotes the lower right and left \(L^{r, \Phi}\)-derivates defined by \[ D_{+, r}\Psi(x;\Phi)=\Big\{\text{ least upper bound of all } \alpha \text{ such that } \] \[ \Big(\int_{a}^{b}[\Psi(x+t)-\Psi(x)-\alpha(\Phi(x+t)-\Phi(x))]^{r}_{-}d t\Big)^{1/r}=o(h)\Big\}, \] and \[ D_{-, r}\Psi(x;\Phi)=\Big\{\text{ least upper bound of all } \alpha \text{ such that } \] \[ \Big(\int_{a}^{b}[-\Psi(x-t)+\Psi(x)-\alpha(-\Phi(x-t)+\Phi(x))]^{r}_{-}d t\Big)^{1/r}=o(h)\Big\}. \] Some properties of the new integral are obtained.