Defining the Mean of a Real-Valued Function on an Arbitrary Metric Space

TL;DR

This paper introduces a way to define a mean functional for real-valued functions on any metric space, leading to a new concept of a relative measure on subsets of the space.

Contribution

It presents a novel method to derive a linear functional (mean) from a metric space, enabling measure definition on arbitrary metric spaces.

Findings

Defines a mean functional for functions on metric spaces

Establishes a new approach to measure subsets relative to the metric space

Provides a framework for analyzing functions and measures in abstract metric settings

Abstract

We show how a metric space induces a linear functional (a "mean") on real-valued functions with domains in that metric space. This immediately induces a "relative" measure on a collection of subsets of the underlying set.