2023/02/15 by S. Okada, Okada, S., J. Rodríguez +3 · 1 citation
Mathematics · #46E30 #46G10 #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.2302.07485
openalex publication_date 2023/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study some aspects of countably additive vector measures with values in ℓ_∞ and the Banach lattices of real-valued functions that are integrable with respect to such a vector measure. On the one hand, we prove that if W ⊆ ℓ_∞^* is a total set not containing sets equivalent to the canonical basis of ℓ1(\mathfrakc), then there is a non-countably additive ℓ_∞-valued map ν defined on a σ-algebra such that the composition x^* ∘ ν is countably additive for every x^*∈ W. On the other hand, we show that a Banach lattice E is separable whenever it admits a countable positively norming set and both E and E^* are order continuous. As a consequence, if ν is a countably additive vector measure defined on a σ-algebra and taking values in a separable Banach space, then the space L1(ν) is separable whenever L1(ν)^* is order continuous.