On the Differentiability of Vector Valued Additive Set Functions

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.

KEYWORDS

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

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

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.

M. Robdera and D. Kagiso, "On the Differentiability of Vector Valued Additive Set Functions,"

References

[1] M. A. Robdera, “Unified Approach to Vector Valued Integration,” International Journal of Functional Analysis, Operator Theory and Application, Vol. 5, No. 2, 2013, pp. 119-139.

[2] S. Bochner, “Additive Set Functions on Groups,” Annals of Mathematics, Second Series, Vol. 40, No. 4, 1939, pp. 769-799.

[3] J. Diestel and J. J. Uhl Jr., “Vector Measures,” American Mathematical Society, Providence, R.I., 1977.

[4] E. J. McShane, “Partial Orderings and Moore-Smith Limits,” American Mathematical Monthly, Vol. 59, No. 1, 1952, pp. 1-11. http://dx.doi.org/10.2307/2307181

[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

[1] M. A. Robdera, “Unified Approach to Vector Valued Integration,” International Journal of Functional Analysis, Operator Theory and Application, Vol. 5, No. 2, 2013, pp. 119-139.

[2] S. Bochner, “Additive Set Functions on Groups,” Annals of Mathematics, Second Series, Vol. 40, No. 4, 1939, pp. 769-799.

[3] J. Diestel and J. J. Uhl Jr., “Vector Measures,” American Mathematical Society, Providence, R.I., 1977.

[4] E. J. McShane, “Partial Orderings and Moore-Smith Limits,” American Mathematical Monthly, Vol. 59, No. 1, 1952, pp. 1-11. http://dx.doi.org/10.2307/2307181

[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