On the differentiability of vector valued additive set functions
Robdera, Mangatiana A.
PublisherScientific Research, http://www.scirp.org
RightsAvailabe under Creative Common Attribution License
Rights holderRobdera, Mangatiana
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The Lebesgue-Nikodým Theorem states that for a Lebesgue measure λ:Σ〖⊂2〗^Ω→[0,∞) an additive set function F:Σ→R which is λ-absolutely continuous is the integral of a Lebegsue integrable a measurable function f:Ω→R; that is, for all measurable sets A, F(A)=∫_A▒〖fdλ.〗 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.