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.
Share and Cite:
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.
Conflicts of Interest
The authors declare no conflicts of interest.
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
|