The Cartan, Choquet and Kellogg properties for the fine topology on metric spaces

TL;DR

This paper establishes fundamental properties of the fine topology on certain metric spaces, including the Cartan, Choquet, and Kellogg properties, and explores implications for finely continuous functions and capacitary measures.

Contribution

It proves the Cartan and Choquet properties for the fine topology on metric spaces with doubling measures and Poincaré inequalities, and applies these to characterize finely continuous functions and capacitary measures.

Findings

Proves Cartan and Choquet properties for the fine topology.

Establishes a fine Kellogg property and characterizes finely continuous functions.

Shows capacitary measures are supported on the fine boundary.

Abstract

We prove the Cartan and Choquet properties for the fine topology on a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, $1 < p< \infty$. We apply these key tools to establish a fine version of the Kellogg property, characterize finely continuous functions by means of quasicontinuous functions, and show that capacitary measures associated with Cheeger supersolutions are supported by the fine boundary of the set.