Regulated functions. *-integral

TL;DR

This book systematically explores the theory of regulated functions, functions of bounded variation, and introduces the *-integral, a new integral concept that simplifies certain limit theorems and applications in function analysis.

Contribution

It introduces the *-integral, a novel integration method for sigma-continuous functions over bounded variation functions, with more convenient limit theorems.

Findings

The *-integral simplifies limit theorems compared to Lebesgue-Stieltjes integral.

Provides detailed analysis of regulated and sigma-continuous functions.

Discusses applications of the *-integral in function theory.

Abstract

The present book gives a systematic overview of function theory and the theory of Stieltjes integral. In particular, we give a detailed account of the theory of functions of bounded variation and of the theory of regulated functions (= functions having finite one-sided limits at each point of their domain). We also present a detailed discussion of $\sigma$-continuous functions (= functions having at most countable set of points of discontinuity) and of the theory of Riemann-Stieltjes integral. We introduce the notion of *-integral, which allows us to integrate $\sigma$-continuous functions over functions of bounded variation. The limit theorems for *-integral are somewhat more convenient than their analogues for the general Lebesgue-Stieltjes integral. Some applications of *-integral are also discussed.