An Overview on the Properties of Vector-Valued Measurable Functions

A Study of Riemann Measurable Vector-Valued Functions and their Properties

Abstract

This paper intends to study the Vector –Valued Measurable functions and its related properties. Study in this paper is not restricted to, but, we do have extensively dealt with the class of ‘Riemann measurable’ vector-valued functions and ‘Lusin type property’. This class contains all Riemann integrable functions and is closely related to the restricted versions of the McShane and Henstock integrals, the M-and H-integrals, defined by means of Lebesgue measurable gauges. Not exclusively but primarily, in this paper, our developments are in the spirit of the Riemann type integral theory for real-valued functions. In particular, we prove that a bounded Riemann measurable vector-valued function is M -integrable.