Let X be a Banach space, 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-3,
author = {J. Rodr\'\i guez and G. Vera},
title = {Uniqueness of measure extensions in Banach spaces},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {139-155},
zbl = {1141.28301},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-3}
}
J. Rodríguez; G. Vera. Uniqueness of measure extensions in Banach spaces. Studia Mathematica, Tome 173 (2006) pp. 139-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm175-2-3/