Uniform structure and topological group properties of non-additive measure algebras

This paper introduces a novel framework for analyzing the uniform structure and topological group properties of measurable spaces with non-additive measures. Unlike traditional approaches that rely on additivity, new concepts, uniformness, K-uniformness, and infra K-uniformness, are defined to establish a pseudo-metric and a topological group structure for non-additive measure algebras. The results demonstrate that these new uniformity conditions allow for a more general treatment of non-additive measures, bridging the gap between classical measure theory and fuzzy measure spaces, focusing on a quasi-monotone measure μ defined on an algebra $A\subset P\left(X\right)$. The space $(A,\mu )$ is an abelian group under the symmetric difference operation. We define the concepts of uniformness, K-uniformness, and infra K-uniformness for μ, establishing conditions under which $(A,\mu )$ becomes a topological group. By constructing a uniform structure and a pseudo-metric on $(A,\mu )$, we demonstrate that if μ is K-uniform, it relates closely to subadditive measures through power transformations.