Advances in Pure Mathematics
Volume 3, Issue 8 (November 2013)
ISSN Print: 2160-0368 ISSN Online: 2160-0384
Google-based Impact Factor: 0.50 Citations h5-index & Ranking
On the Differentiability of Vector Valued Additive Set Functions ()
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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.
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