Multipliers of vector valued McShane integrable functions in locally convex space

Abstract

We study measurable real valued multipliers of variationally~\linebreak McShane (resp. McShane) integrable functions defined on a $\sigma$-finite outer regular quasi-Radon measure space and taking values in a complete locally convex topological vector space~$X$.
 We also show: in case $X$ is representable by semi-norm then essentially bounded real measurable functions  are multipliers of functions which are Pettis integrable as well as integrable by semi-norm.
 The space of real valued measurable and essentially bounded functions turn out to be precisely the  multipliers of variationally McShane
 (resp. McShane) integrable functions in~case~$X$
 is representable
 by~semi-norm.