# Positive Representations of L 1 of a Vector Measure

Positive Representations of L 1 of a Vector Measure We characterize the vector measures n on a Banach lattice such that the map $$\|\int|\cdot|dn \|$$ provides a quasi-norm which is equivalent to the canonical norm $$\|\cdot\|_{n}$$ of the space L 1(n) of integrable functions as an specific type of transformations of positive vector measures that we call cone-open transformations. We also prove that a vector measure m on a Banach space X constructed as a cone-open transformation of a positive vector measure can be considered in some sense as a positive vector measure by defining a new order on X. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Positive Representations of L 1 of a Vector Measure

Positivity, Volume 11 (3) – Jan 1, 2007
11 pages

/lp/springer_journal/positive-representations-of-l-1-of-a-vector-measure-TGJey8WSNF
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2075-9
Publisher site
See Article on Publisher Site

### Abstract

We characterize the vector measures n on a Banach lattice such that the map $$\|\int|\cdot|dn \|$$ provides a quasi-norm which is equivalent to the canonical norm $$\|\cdot\|_{n}$$ of the space L 1(n) of integrable functions as an specific type of transformations of positive vector measures that we call cone-open transformations. We also prove that a vector measure m on a Banach space X constructed as a cone-open transformation of a positive vector measure can be considered in some sense as a positive vector measure by defining a new order on X.

### Journal

PositivitySpringer Journals

Published: Jan 1, 2007

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