Local Radon-Nikodym Derivatives of Set Functions

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Proceedings of the American Mathematical Society, 1975

The Radon-Nikodym theorems of Segal and Zaanen are principally concerned with the classification of those measures p. for which any X« p. is given in the form (i) KA) = fAgdf or all sets A of finite p. measure. This paper is concerned with the characterization of those pairs X , p. for which the equality (i) holds for every measurable set A, and introduces a notion of compatibility that essentially solves this problem. In addition, some applications are made to Radon-Nikodym theorems for regular Borel measures.

We extend additive set-valued set functions and normal multimeasures defined on a ring of subsets. We also prove a Carathéodory-Hahn-Kluvanek-type theorem for additive set-valued set functions. Finally, we establish results on the extension of transition multimeasures.

Proceedings of the Edinburgh Mathematical Society, 1978

Let (S, ℳ) be a measurable space (that is, a set S in which is defined a σ-algebra ℳ of subsets) and X a locally convex space. A map M from ℳ to the family of all non-empty subsets of X is called a multimeasure iff for every sequence of disjoint sets An ɛ ℳ (n=1,2,… )with the series converges (in the sense of (6), p. 3) to M(A).

Bulletin of the Australian Mathematical Society, 2000

We prove that a Banach space X has the Radon-Nikodym property if, and only if, every weak*-lower semicontinuous convex continuous function f of X* is Gâteaux differentiable at some point of its domain with derivative in the predual space x.