Essential Closures

Abstract

We propose the notion of general essential closures in an axiomatic way and develop the theory of essential closures. In the process, we introduce the notions of essential closedness and essential compactness analogous to the notions of topological closedness and topological compactness, respectively. We also introduce the notion of essentiality which is the core of the theory of essential closures. It is what distinguishes the notion of essential closures from the notion of topological closures. For applications, we construct concrete examples of essential closures via the notion of submeasures, which are restrictions of outer measures, and use them as tools to study supports of absolutely continuous measures and supports of multivariate copulas, and to introduce the notion of essential supports of functions.