UBC Theses and Dissertations
Differentiation of group-valued outer measures Traynor , Tim Eden
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.
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