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Differentiation of group-valued outer measures

University of British Columbia

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.

Text

2011-06-23

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http://hdl.handle.net/2429/35688

Mathematics

University of British Columbia

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