On the existence of the Stieltjes integral

H. L. Smith
1925 Transactions of the American Mathematical Society  
Lebesguet 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
more » ... ontinuous) 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.
doi:10.1090/s0002-9947-1925-1501324-5 fatcat:vvd4sok445hgvffvt3w25sfhwa