µ-Integrable Functions and Weak Convergence of Probability Measures in Complete Paranormed Spaces

Renying Zeng ()
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Renying Zeng: Mathematics Department, Saskatchewan Polytechnic, Saskatoon, SK S7L 4J7, Canada

Mathematics, 2024, vol. 12, issue 9, 1-10

Abstract: This paper works with functions defined in metric spaces and takes values in complete paranormed vector spaces or in Banach spaces, and proves some necessary and sufficient conditions for weak convergence of probability measures. Our main result is as follows: Let X be a complete paranormed vector space and Ω an arbitrary metric space, then a sequence { μ n } of probability measures is weakly convergent to a probability measure μ if and only if lim n → ∞ ∫ Ω g ( s ) d μ n = ∫ Ω g ( s ) d μ for every bounded continuous function g : Ω → X . A special case is as the following: if X is a Banach space, Ω an arbitrary metric space, then { μ n } is weakly convergent to μ if and only if lim n → ∞ ∫ Ω g ( s ) d μ n = ∫ Ω g ( s ) d μ for every bounded continuous function g : Ω → X . Our theorems and corollaries in the article modified or generalized some recent results regarding the convergence of sequences of measures.

Keywords: finite measure; Banach space; complete paranormed space; ?-integral function; weak convergence of measures (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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