# Derivability, variation and range of a vector measure

Studia Mathematica (1995)

• Volume: 112, Issue: 2, page 165-187
• ISSN: 0039-3223

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## Abstract

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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 that the range of a measure determines its total variation. We also give a new proof of this fact.

## How to cite

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Rodríguez-Piazza, L.. "Derivability, variation and range of a vector measure." Studia Mathematica 112.2 (1995): 165-187. <http://eudml.org/doc/216144>.

@article{Rodríguez1995,
abstract = {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 that the range of a measure determines its total variation. We also give a new proof of this fact.},
author = {Rodríguez-Piazza, L.},
journal = {Studia Mathematica},
keywords = {vector measures; range; variation; Bochner derivability; zonoid; range of a vector measure; -finiteness of its variation; derivability of the measure; Bochner derivative},
language = {eng},
number = {2},
pages = {165-187},
title = {Derivability, variation and range of a vector measure},
url = {http://eudml.org/doc/216144},
volume = {112},
year = {1995},
}

TY - JOUR
AU - Rodríguez-Piazza, L.
TI - Derivability, variation and range of a vector measure
JO - Studia Mathematica
PY - 1995
VL - 112
IS - 2
SP - 165
EP - 187
AB - 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 that the range of a measure determines its total variation. We also give a new proof of this fact.
LA - eng
KW - vector measures; range; variation; Bochner derivability; zonoid; range of a vector measure; -finiteness of its variation; derivability of the measure; Bochner derivative
UR - http://eudml.org/doc/216144
ER -

## References

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