Uniqueness of measure extensions in Banach spaces

• Autorzy: J. Rodríguez , G. Vera
• Języki publikacji: EN

Abstrakty

EN

Let X be a Banach space, $B ⊂ B_{X*}$ a norming set and 𝔗(X,B) the topology on X of pointwise convergence on B. We study the following question: given two (non-negative, countably additive and finite) measures μ₁ and μ₂ on Baire(X,w) which coincide on Baire(X,𝔗(X,B)), does it follow that μ₁ = μ₂? It turns out that this is not true in general, although the answer is affirmative provided that both μ₁ and μ₂ are convexly τ-additive (e.g. when X has the Pettis Integral Property). For a Banach space Y not containing isomorphic copies of ℓ¹, we show that Y* has the Pettis Integral Property if and only if every measure on Baire(Y*,w*) admits a unique extension to Baire(Y*,w). We also discuss the coincidence of the two σ-algebras involved in such results. Some other applications are given.

Kategorie tematyczne

• 28A20: Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
• 28C15: Set functions and measures on topological spaces (regularity of measures, etc.)
• 28B05: Vector-valued set functions, measures and integrals
• 46B26: Nonseparable Banach spaces
• 46G10: Vector-valued measures and integration

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2006
• Tom: 175
• Numer: 2
• Strony: 139-155

Daty

wydano

2006

Twórcy

autor

J. Rodríguez

• Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain

autor

G. Vera

• Departamento de Matemáticas, Universidad de Murcia, 30100 Espinardo (Murcia), Spain

Identyfikatory

DOI

10.4064/sm175-2-3