TITLE:
On the Differentiability of Vector Valued Additive Set Functions
AUTHORS:
Mangatiana A. Robdera, Dintle Kagiso
KEYWORDS:
Vector Valued Additive Set Function; Lebesgue-Radon-Nikodým Theorem; Fundamental Theorem of Calculus
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.3 No.8,
November
27,
2013
ABSTRACT: The Lebesgue-Nikodym Theorem states that
for a Lebesgue measure an additive set function which is -absolutely continuous is the
integral of a Lebegsue integrable a measurable function ; that is, for all measurable
sets.Such a property is not shared by vector valued
set functions. We introduce a suitable definition of the integral that will
extend the above property to the vector valued case in its full generality. We
also discuss a further extension of the Fundamental Theorem of Calculus for
additive set functions with values in an infinite dimensional normed space.