McShane Integrability Using Variational Measure

Felipe Raypan Sumalpong Jr, Julius V. Benitez

Abstract

If f : [a, b] → R is McShane integrable on [a, b], then f is McShane integrable on every Lebesgue measurable subset of [a, b]. However, integrability of a real-valued function on [a, b] does not imply McShane integrability on any E ⊆ [a, b]. In this paper, we give a characterization for the McShane integrability of f : [a, b] → R over E ⊆ [a, b] using concept of variational measure.

Keywords

McShane integral, integrable set, McShane $\delta$-variation, McShane variational measure, variation zero, Cauchy extension

Full Text:

PDF