论文标题
Harnack扩展原理在Kurzweil-Stieltjes积分中的作用
Role of the Harnack Extension Principle in the Kurzweil-Stieltjes Integral
论文作者
论文摘要
使用量规集成的各种stieltjes积分已在微分方程和其他应用领域非常流行。在集成和普通微分方程的理论中，收敛定理提供了最广泛使用的工具之一。 Harnack扩展原理讨论了在$（a，b）$的特定子集上可在$ [a，b] $上集成的特定子集上的库兹维尔·亨斯托克整合功能的足够条件，这是提供收敛定理的关键步骤。每当积分器是身份函数时，Kurzweil-Stieltjes积分就会减少到Kurzweil-Henstock积分。通常，如果集成商$ f $在$ [c，d] \ subset [a，b] $上不连续，那么kurzweil-stieltjes积分的值$$ \ int_c^d [df] g，\ \ \ \ \ \ \ \ int _ {[c，d] [c，d]} [c，d]} \ int _ {（c，d]} [df] g，\ {\ rm and} \ \ \ \ \ int _ {（c，c，d）} [df] g $ g $不需要重合。因此，harnack扩展原理在kurzweil-henstock Integral中不再有效地与kurzweil-seters组成，因为将其组成一致。平等性和等效性的概念对于kurzweil-stieltjes整合的Harnack扩展原理的概念至关重要。 此外，积分$ \ int_a^b [df] g $的存在（即使在身份集成器的情况下）总是暗示每个子集$ t $ t $ of $ $ t $ of $ [a，b] $的积分$ \ int_ {t} [df] g $。这是从一个众所周知的事实来看，如果例如$ t \ subset [a，b] $是不可测量的，那么lebesgue积分$ \ int_a^b g [dt] $的存在并不意味着存在积分$ \ int_t g [dt] $。因此，除了为kurzweil-stieltjes积分构建harnack扩展原理外，本文的目的还旨在证明其在保证积分$ \ int_ {t} [t} [df] g $的存在方面的作用，用于任意子集$ t $ t $ t $ a $ t $。
Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence theorems provide one of the most widely used tools. The Harnack extension principle, which discusses a sufficient condition for Kurzweil-Henstock integrable functions on particular subsets of $(a,b)$ to be integrable on $[a,b]$, is a key step to supply convergence theorems. The Kurzweil-Stieltjes integral reduces to the Kurzweil-Henstock integral whenever the integrator is an identity function. In general, if the integrator $F$ is discontinuous on $[c,d]\subset[a,b]$, then the values of the Kurzweil-Stieltjes integrals $$\int_c^d[dF]g,\ \int_{[c,d]}[dF]g,\ \int_{[c,d)}[dF]g,\ \int_{(c,d]}[dF]g,\ {\rm and}\ \int_{(c,d)}[dF]g$$ need not coincide. Hence, the Harnack extension principle in the Kurzweil-Henstock integral cannot be valid any longer for the Kurzweil-type Stieltjes integrals with discontinuous integrators. The new concepts of equi-integrability and equiregulatedness are pivotal to the notion of the Harnack extension principle for the Kurzweil-Stieltjes integration. Moreover, the existence of the integral $\int_a^b[dF]g$ does not (even in the case of the identity integrator) always imply the existence of the integral $\int_{T}[dF]g$ for every subset $T$ of $[a,b]$. This follows from the well-known fact that, if e.g., $T\subset[a,b]$ is not measurable, then the existence of the Lebesgue integral $\int_a^b g [dt]$ does not imply that the integral $\int_T g [dt]$ exists. Therefore, besides constructing the Harnack extension principle for the abstract Kurzweil-Stieltjes integral, the aim of this paper is also to demonstrate its role in guaranteeing the existence of the integrals $\int_{T}[dF]g$ for arbitrary subsets $T$ of an elementary set $E$.
