# Vector measures: where are their integrals?

Vector measures: where are their integrals? Let ν be a vector measure with values in a Banach space Z. The integration map $$I_\nu: L^1(\nu)\to Z$$ , given by $$f\mapsto \int f\,d\nu$$ for f ∈ L 1(ν), always has a formal extension to its bidual operator $$I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$$ . So, we may consider the “integral” of any element f ** of L 1(ν)** as I ν ** (f **). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z **. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I ν ** for the particular vector measure ν defined by ν(A) := T(χ A ). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Vector measures: where are their integrals?

, Volume 13 (1) – Apr 5, 2008
27 pages

/lp/springer_journal/vector-measures-where-are-their-integrals-tSxa0FOQ7v
Publisher
Birkhäuser-Verlag
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/s11117-008-2191-1
Publisher site
See Article on Publisher Site

### Abstract

Let ν be a vector measure with values in a Banach space Z. The integration map $$I_\nu: L^1(\nu)\to Z$$ , given by $$f\mapsto \int f\,d\nu$$ for f ∈ L 1(ν), always has a formal extension to its bidual operator $$I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$$ . So, we may consider the “integral” of any element f ** of L 1(ν)** as I ν ** (f **). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z **. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I ν ** for the particular vector measure ν defined by ν(A) := T(χ A ).

### Journal

PositivitySpringer Journals

Published: Apr 5, 2008

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