Applications of MR-Metric Spaces in Measure Theory and Convergence Analysis

Abstract

This paper investigates the role of MR-metric spaces in the fields of measure theory and convergence analysis. We analyze how measures defined through a triadic metric function reveal key characteristics such as $\sigma$-finiteness and absolute continuity. Furthermore, we explore the convergence behavior of sequences within the MR-metric framework, highlighting their relevance in areas like stochastic processes, optimization, and data science. The findings offer valuable perspectives on probability distributions that incorporate triadic dependencies, the conditions necessary for stability in machine learning, and the clustering behavior within complex networks. These discoveries pave the way for extending traditional metric space theory into more advanced settings, including those involving multi-agent systems and high-dimensional analyses.