On the Existence of the Stieltjes Integral
1925; American Mathematical Society; Volume: 27; Issue: 4 Linguagem: Inglês
10.2307/1989238
ISSN1088-6850
Autores Tópico(s)Nonlinear Differential Equations Analysis
ResumoLebesguet has shown how to treat the Riemann integral by studying the content of an associated point set.The present paper is an attempt in the same direction for the Stieltjes integral.A pair of conditions are found which are necessary and sufficient for the existence of the integral, one of which concerns itself with the associated point set.The other is automatically satisfied for a large class of integrals comprising (1) those for which the associated point set is a (continuous) curve with at most a finite number of multiple points; and (2) those for which the measure function is of limited variation.A consequence is that a simple closed curve must be squarable if its line integral J y dx exists.Among the examples given is one which shows that a simple closed curve may be squarable and still fail to have an existent line integral J y dx.1. Definitions and notations.If T, I" are two sub -intervals of T, T-I" will denote the interval common to P, I".A general partition to -0 < tx < ■ • • < tn-x <tn = 1 of the interval T: 0 < t < 1 will be denoted by the notation n; a partition of a sub-
Referência(s)