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Decomposition of group-valued additive set functions

Tim Traynor (1972)

Annales de l'institut Fourier

Let m be an additive function on a ring H of sets, with values in a commutative Hausdorff topological group, and let K be an ideal of H . Conditions are given under which m can be represented as the sum of two additive functions, one essentially supported on K , the other vanishing on K . The result is used to obtain two Lebesgue-type decomposition theorems. Other applications and the corresponding theory for outer measures are also indicated.

Denseness and Borel complexity of some sets of vector measures

Zbigniew Lipecki (2004)

Studia Mathematica

Let ν be a positive measure on a σ-algebra Σ of subsets of some set and let X be a Banach space. Denote by ca(Σ,X) the Banach space of X-valued measures on Σ, equipped with the uniform norm, and by ca(Σ,ν,X) its closed subspace consisting of those measures which vanish at every ν-null set. We are concerned with the subsets ν ( X ) and ν ( X ) of ca(Σ,X) defined by the conditions |φ| = ν and |φ| ≥ ν, respectively, where |φ| stands for the variation of φ ∈ ca(Σ,X). We establish necessary and sufficient conditions...

Derivability, variation and range of a vector measure

L. Rodríguez-Piazza (1995)

Studia Mathematica

We prove that the range of a vector measure determines the σ-finiteness of its variation and the derivability of the measure. Let F and G be two countably additive measures with values in a Banach space such that the closed convex hull of the range of F is a translate of the closed convex hull of the range of G; then F has a σ-finite variation if and only if G does, and F has a Bochner derivative with respect to its variation if and only if G does. This complements a result of [Ro] where we proved...

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