  We propose a new approach to vague convergence of measures based on the
general theory of boundedness due to Hu (1966). The article explains how this
connects and unifies several frequently used types of vague convergence from
the literature. Such an approach allows one to translate already developed
results from one type of vague convergence to another. We further analyze the
corresponding notion of vague topology and give a new and useful
characterization of convergence in distribution of random measures in this
topology.
