UNCERTAINTY-AWARE EXTENSIONS OF RADON NIKODÝM AND HAHN BANACH THEOREMS IN NEUTROSOPHIC MR-METRIC SPACES

Abstract

This paper introduces and systematically investigates the concept of Neutro-sophic MR-Metric Spaces(NMR-MS), a novel hybrid structure that combines the exibilityof MR-metric spaces with the expressive power of neutrosophic logic. By incorporating thethree independent neutrosophic membership degrees|truth (𝓣), indeterminacy (𝓘), andfalsity (𝓕)-into the MR-metric framework, we develop a robust mathematical environmentcapable of modeling uncertainty, indeterminacy, and conicting information. Within thissetting, two foundational theorems are established: the Neutrosophic MR-Radon-NikodymTheorem, which provides a decomposition of measures into classical and neutrosophic com-ponents, and the Neutrosophic Hahn{Banach Extension Theorem, which extends linear func-tionals while preserving neutrosophic structure. The theoretical results are accompanied bya wide range of applications in measure theory, functional analysis, signal processing, financial risk assessment, and machine learning. This work bridges classical analytical methodswith uncertainty-aware models, oering powerful tools for modern applied mathematics anddecision-making under indeterminacy.