A Note on Measures Determined by Continuous Functions

Abstract

Ellis and Jeffery [2] studied Borel measures determined in a certain way by real valued functions of a real variable which have finite left and right hand limits at each point. If f is such a function and is of bounded variation on an interval I, then the associated measure μf has the property that μf(I) equals the total variation of f on I. The authors then indicated in [3] how some of these measures permit the definition of generalized integrals of Denjoy type. In [1], the authors construct an example of a continuous function f, not of bounded variation, such that the associated measure μf is the zero measure. The purpose of this note is to show that "most" continuous functions give rise to the zero measure in the sense that there is a residual subset R of C[a, b] such that for each f∊R, the associated measure μf is the zero measure.