A functional characterization of measures and the Banach–Ulam problem

For a measurable space ( Ω , A ) , let ℓ ∞ ( A ) be the closure of span { χ A : 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 ϕ μ defined by 〈 ϕ μ , x 〉 = ∫ Ω x d μ (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 ℓ 1 -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.