Fine properties and a notion of quasicontinuity for BV functions on metric spaces

TL;DR

This paper investigates the fine properties of BV functions on metric spaces with doubling measures and Poincaré inequalities, establishing a form of quasicontinuity outside jump sets and implications for sets of finite perimeter.

Contribution

It introduces a notion of quasicontinuity for BV functions on metric spaces and links measure-theoretic boundaries with topological and curve-based separation properties.

Findings

BV functions are continuous outside jump sets after removing small capacity sets

Measure-theoretic boundary separates interior and exterior in topology and curve sense

Establishes a quasicontinuity property for BV functions in metric spaces

Abstract

On a metric space equipped with a doubling measure supporting a Poincar\'e inequality, we show that given a BV function, discarding a set of small $1$-capacity makes the function continuous outside its jump set and ``one-sidedly" continuous in its jump set. We show that such a property implies, in particular, that the measure theoretic boundary of a set of finite perimeter separates the measure theoretic interior of the set from its measure theoretic exterior, both in the sense of the subspace topology outside sets of small $1$-capacity, and in the sense of $1$-almost every curve.