Abdulaziz Alo - Academia.edu (original) (raw)

Papers by Abdulaziz Alo

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 ... more 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.

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 ... more 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.