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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
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.
2
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Compactness in L¹ of a vector measure

81%
Calabuig J., Lajara S., Rodríguez J., Sánchez-Pérez E.
Studia Mathematica
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2014
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tom 225
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nr 3
259-282
EN
We study compactness and related topological properties in the space L¹(m) of a Banach space valued measure m when the natural topologies associated to convergence of vector valued integrals are considered. The resulting topological spaces are shown to be angelic and the relationship of compactness and equi-integrability is explored. A natural norming subset of the dual unit ball of L¹(m) appears in our discussion and we study when it is a boundary. The (almost) complete continuity of the integration operator is analyzed in relation with the positive Schur property of L¹(m). The strong weakly compact generation of L¹(m) is discussed as well.
3
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An algebraic derivative associated to the operator $D^δ$

81%
Almeida V. M., Castro N., Rodríguez J.
Banach Center Publications
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2000
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tom 53
|
nr 1
71-78
EN
In this paper we get an algebraic derivative relative to the convolution $(f*g)(t)=∫_0^ti f(t-ψ)g(ψ)dψ$ associated to the operator $D^δ$, which is used, together with the corresponding operational calculus, to solve an integral-differential equation. Moreover we show a certain convolution property for the solution of that equation
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