Uniqueness of measure extensions in Banach spaces

J. Rodríguez, and G. Vera. "Uniqueness of measure extensions in Banach spaces." Studia Mathematica 175.2 (2006): 139-155. <http://eudml.org/doc/286091>.

@article{J2006,
abstract = {Let X be a Banach space, $B ⊂ B_\{X*\}$ a norming set and (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,(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.},
author = {J. Rodríguez, G. Vera},
journal = {Studia Mathematica},
keywords = {measure extension; Baire measure; Banach space; Pettis integral},
language = {eng},
number = {2},
pages = {139-155},
title = {Uniqueness of measure extensions in Banach spaces},
url = {http://eudml.org/doc/286091},
volume = {175},
year = {2006},
}

TY - JOUR
AU - J. Rodríguez
AU - G. Vera
TI - Uniqueness of measure extensions in Banach spaces
JO - Studia Mathematica
PY - 2006
VL - 175
IS - 2
SP - 139
EP - 155
AB - Let X be a Banach space, $B ⊂ B_{X*}$ a norming set and (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,(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.
LA - eng
KW - measure extension; Baire measure; Banach space; Pettis integral
UR - http://eudml.org/doc/286091
ER -