Effective convergence in computable measure theory

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2022-05
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Rojas, Diego Antonio
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McNicholl, Timothy H
Lutz, Jack H
Weber, Eric
Herzog, David P
Martin, Ryan
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Altmetrics
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Mathematics
Abstract
In this dissertation, we investigate effective notions of weak and vague convergence of measures on R. First, we describe the framework to study the effective theory of weak convergence of measures on R introduced in [22] and continued in [27]. We give two effective definitions of weak convergence: one uniform and one non-uniform. Although the uniform definition is strictly stronger than the non-uniform definition, we show that they are equivalent for computable sequences of measures. Moreover, we exhibit effective versions of key theorems in the theory of weak convergence of measures: one attributed to Alexandroff [1] and another due to Prokhorov [25]. Next, we describe the framework to study the effective theory of vague convergence of measures on R introduced in [27]. As with effective weak convergence, we give a uniform and a non-uniform effective definition of vague convergence and show that these two definitions are equivalent for computable sequences of measures. Whereas effective weak limits are guaranteed to be computable, effective vague limits may not be computable. We give an example of a computable sequence of measures that effectively vaguely converges to a finite, incomputable measure, and we give a necessary and sufficient condition for an effective vague limit measure to be computable. Furthermore, we determine a condition which establishes a correspondence between effective weak and vague convergence of measures.
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