Median pretrees and functions of bounded variation

TL;DR

This paper introduces functions of bounded variation on median algebras and pretrees, establishing properties like the point of continuity and a generalized Helly's theorem for such functions on compact metrizable median pretrees.

Contribution

It extends the concept of bounded variation to median algebras and proves new properties and a Helly-type theorem in this setting.

Findings

Functions of bounded variation on median pretrees have the point of continuity property.

A generalized Helly's selection theorem is proved for sequences of functions with bounded variation.

Results apply to compact metrizable median pretrees in their shadow topology.

Abstract

We introduce functions of bounded variation on median algebras and study some properties for median pretrees. We show that if $X$ is a compact median pretree in its shadow topology then every function $f: X \to R$ of bounded variation has the point of continuity property (Baire 1, if $X$, in addition, is metrizable). We prove a generalized version of Helly's selection theorem for a sequence of functions with total bounded variation defined on a compact metrizable median pretree $X$.