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On the Differentiability of Vector Valued Additive Set Functions

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DOI: 10.4236/apm.2013.38087    2,117 Downloads   3,736 Views   Citations

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Robdera and D. Kagiso, "On the Differentiability of Vector Valued Additive Set Functions," Advances in Pure Mathematics, Vol. 3 No. 8, 2013, pp. 653-659. doi: 10.4236/apm.2013.38087.

References

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[5] M. A. Robdera, “On Strong and Weak Integrability of Vector Valued Functions,” International Journal of Functional Analysis, Operator Theory and Application, Vol. 5, No. 1, 2013, pp. 63-81.
[6] J. T. Lu and P. Y. Pee, “The Primitive of a Henstock Integrable Functions in Euclidean Space,” Bulletin of the London Mathematical Society, Vol. 31, No. 2, 1999, pp. 173-180. http://dx.doi.org/10.1112/S0024609398005347

  
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