The Lebesgue-Nikodým Theorem states that for a Lebesgue measure \lambda:\Sigma\subset2^{\Omega}\rightarrow[0,\infty), an additive set function F:\Sigma\rightarrow\mathbb{R} which is \lambda-absolutely continuous is the integral of a Lebegsue integrable a measurable function f:\Omega\rightarrow\mathbb{R}; that is, for all measurable sets A, F(A)=\int_{A}fd\lambda. 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.

Keywords: Vector Valued Additive Set Function; Lebesgue-Radon-Nikodým Theorem; Fundamental Theorem of Calculus

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