Correction of metrics

TL;DR

This paper proves that symmetric nonnegative functions satisfying the triangle inequality almost everywhere are equivalent to semimetrics, and discusses properties of metric triples in measure and metric spaces.

Contribution

It establishes the equivalence between certain functions and semimetrics under almost everywhere conditions, advancing understanding of metric structures in measure spaces.

Findings

Symmetric nonnegative functions satisfying the triangle inequality are equivalent to semimetrics.

Properties of metric triples in measure and metric spaces are analyzed.

The paper provides foundational results linking functions and metric structures.

Abstract

We prove that a symmetric nonnegative function of two variables on a Lebesgue space that satisfies the triangle inequality for almost all triples of points is equivalent to some semimetric. Some other properties of metric triples (spaces with structures of a measure space and a metric space) are discussed.