A functional characterization of measures and the Banach–Ulam problem

For a measurable space (Ω,A), let ℓ∞(A) be the closure of span{χA:A∈A} in ℓ∞(Ω). In this paper we show that a sufficient and necessary condition for a real-valued finitely additive measure μ on (Ω,A) to be countably additive is that the corresponding functional ϕμ defined by (for x∈ℓ∞(A)) is w*-sequentially continuous. With help of the Yosida–Hewitt decomposition theorem of finitely additive measures, we show consequently that every continuous functional on ℓ∞(A) can be uniquely decomposed into the ℓ1-sum of a w*-continuous functional, a purely w*-sequentially continuous functional and a purely (strongly) continuous functional. Moreover, several applications of the results to measure extension are given.