2022/06/08 by Jerzy Kąkol, Kąkol, Jerzy, Damian Sobota +3
Economics, Econometrics and Finance · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Mathematical and Theoretical Analysis #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2206.03794
openalex publication_date 2022/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using elementary probabilistic methods, in particular a variant of the Weak Law of Large Numbers related to the Bernoulli distribution, we prove that for every infinite compact spaces K and L the product K× L admits a sequence ⟨μn\colon n∈ℕ⟩ of normalized signed measures with finite supports which converges to 0 with respect to the weak* topology of the dual Banach space C(K× L)^*. Our approach is completely constructive -- the measures μn are defined by an explicit simple formula. We also show that this result generalizes the classical theorem of Cembranos and Freniche which states that for every infinite compact spaces K and L the Banach space C(K× L) contains a complemented copy of the space c0.