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Uniqueness of measure extensions in Banach spaces

100%

Rodríguez J., Vera G.

Studia Mathematica

2006

tom 175

nr 2

139-155

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.

Fragmentability and compactness in C(K)-spaces

81%

Cascales B., Manjabacas G., Vera G.

Studia Mathematica

1998

tom 131

nr 1

73-87

Let K be a compact Hausdorff space, $C_p(K)$ the space of continuous functions on K endowed with the pointwise convergence topology, D ⊂ K a dense subset and $t_p(D)$ the topology in C(K) of pointwise convergence on D. It is proved that when $C_p(K)$ is Lindelöf the $t_p(D)$-compact subsets of C(K) are fragmented by the supremum norm of C(K). As a consequence we obtain some Namioka type results and apply them to prove that if K is separable and $C_p(K)$ is Lindelöf, then K is metrizable if, and only if, there is a countable and dense subset D ⊂ K such that $(C(K),t_p(D))$ is analytic. We also show that if K is a separable Rosenthal compact space, then K is metrizable if, and only if, $C_p(K)$ is Lindelöf. We complete our study by showing that if K does not contain a copy of βℕ, then convex $t_p(D)$-compact subsets of C(K) have the weak Radon-Nikodym property.

Ograniczanie wyników

2 Studia Mathematica

2 Vera G.

1 Cascales B.

1 Manjabacas G.

1 Rodríguez J.

1 2006

1 1998

Wyniki wyszukiwania

Uniqueness of measure extensions in Banach spaces

Fragmentability and compactness in C(K)-spaces