2019/02/22 by Svetlana V. Butler, Butler, Svetlana V.
Mathematics · #28A25 #28C05 #28C15 #46F99 #46T99 #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results
paper · pdf · doi:10.48550/arxiv.1902.08372
openalex publication_date 2019/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Topological measures and deficient topological measures generalize Borel\nmeasures and correspond to certain non-linear functionals. We study integration\nwith respect to deficient topological measures on locally compact spaces. Such\nan integration over sets yields a new deficient topological measure if we\nintegrate a nonnegative vanishing at infinity function; and it produces a\nsigned deficient topological measure if we use a continuous function on a\ncompact space. We present many properties of these resulting deficient\ntopological measures and of signed deficient topological measures. In\nparticular, they are absolutely continuous with respect to the original\ndeficient topological measure and Lipschitz continuous. Deficient topological\nmeasures obtained by integration over sets can also be obtained from non-linear\nfunctionals. We show that for a deficient topological measure \μ that\nassumes finitely many values, there is a function f such that \∫X f ,\nd \μ = 0, but \∫X (-f ) , d \μ \≠ 0. We present different criteria\nfor \∫X f , d \μ = 0. We also prove some convergence results, including\na Monotone convergence theorem.\n