A note on vague convergence of measures

Bojan Basrak, Hrvoje Planinić

Published 2018-03-19Version 1

We propose a notion of convergence of measures with intention of generalizing and unifying several frequently used types of vague convergence. We explain that by general theory of boundedness due to Hu (1966), in Polish spaces, this notion of convergence can be always formulated as follows: $\mu_n \stackrel{v}{\longrightarrow} \mu$ if $\int f d\mu_n \to \int f d\mu$ for all continuous bounded functions $f$ with support bounded in some suitably chosen metric. This brings all the related types of vague convergence into the framework of Daley and Vere-Jones (2003) and Kallenberg (2017). In the rest of the note we discuss the vague topology and the corresponding notion of convergence in distribution, complementing the theory developed in those two references.