Effective notions of weak convergence of measures on the real line

Effective convergence in computable measure theory

“…Proposition 3.1.2 (McNicholl and R. [22]). Suppose there is a computable witness that lim inf n a n is not smaller than a, and suppose there is a computable witness that lim sup n a n is not larger than a.…”

“…In Chapter 3, we introduce some preliminary material needed to understand the results from Chapter 4. In Chapter 4, we define the effective notions of weak convergence of measures on R introduced in [22], and we present related results from [22] and [27].…”

“…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.…”

“…In this dissertation, we detail the work of McNicholl and D. Rojas [22,27] in which they develop a framework to study the effective theory of weak and vague convergence of measures on the real line R. We formalize what it means for a sequence of measures in M(R) to effectively weakly or effectively vaguely converge in two seemingly distinct ways: non-uniformly and uniformly. In the case of computable sequences of measures, we show that the non-uniform and uniform versions of each effective convergence notion are actually equivalent.…”

“…Lemma 3.2.5 (McNicholl and R. [22]). Suppose f : (0, ∞) → [0, ∞) is an upper semi-computable function whose zero set is dense in (0, ∞).…”