Record number :
1561476
Title of article :
A functional characterization of measures and the Banach–Ulam problem
Author/Authors :
Cheng، نويسنده , , Cheng Lixin and Shi Shuzhong، نويسنده , , Huihua، نويسنده ,
Issue Information :
دوهفته نامه با شماره پیاپی سال 2011
Pages :
8
From page :
558
To page :
565
Abstract :
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.
Keywords :
linear functional , Real-valued measure , Space of measures , representation , EXTENSION , decomposition , w ? -Sequential continuity
Journal title :
Journal of Mathematical Analysis and Applications
Journal title :
Journal of Mathematical Analysis and Applications
Serial Year :
2011
Link To Document :
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