Thesis--University of Florida. / Description based on print version record. Manuscript copy. Vita. Bibliography: leaves 99-100.

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Afhandling--Copenhagen. / Summary in Danish. Bibliography: p. 61-65.

Thesis (Ph. D.)--University of Wisconsin, 1964. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references.

A. Appert proved in [2] that every sequence of strong measures on a separable weakly locally compact metrizable space has a subsequence converging in the sense of Δ to a strong measure. We extend this result to a second countable locally compact regular space. A. Appert also proved that on a weakly locally compact metrizable space, a sequence of strong measures converges in the sense of Δ to a strong measure if it converges in the sense of Δ₁to that strong measure. In [3], D. J. H. Garling extended this result to a sequence of monotone set functions on a weakly locally compact Hausdorff space under certain conditions. We show that this result still holds on a locally compact regular space with the same conditions. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis is divided into three parts. In Part I, we define and give properties of semigroup valued measures and of the indefinite integral ʃ₍ ₎f‧dµ , where f is a many-valued function (i.e. relation) with values In a group µ is an outer measure with values in another group, and "•" is an operation from the cartesian product of the two groups into a third. In particular, we present a Lebesgue decomposition for group-valued outer measures and show that the indefinite integral is an outer measure. In Part II, we construct the many-valued (outer) derivative D̅ of an outer measure Ʋ with respect to the outer measure µ based on the notion of the limit of "approximate ratios" Ʋ(A) to µ(A) as the set A shrinks to a point. D̅ depends upon the multiplication "•" , upon an auxiliary "remainder" function r , and upon the specification of convergence (formula omitted) of sets in the measure space. Conditions are given under which Ʋ =ʃ₍ ₎D̅‧dµ. We also provide a general Hahn decomposition theorem and discuss generalized types of Vitali differentiation systems. In Part III, we give some applications, including Radón-Nikodym theorems for outer measures Ʋ and µ: firstly, when µ has values in the non-negative reals and Ʋ has values in a locally convex space and secondly, when Ʋ and µ have values in a Banach space. / Science, Faculty of / Mathematics, Department of / Graduate

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