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Annihilating a subspace of $L_1$ with the sign of a continuous function

Author(s): Daniel Wulbert
Journal: Proc. Amer. Math. Soc. 128 (2000), 2431-2438.
MSC (1991): Primary 46E30; Secondary 46G10, 26A15
Posted: November 24, 1999
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Abstract: Let $(X,\Sigma, \mu)$ be a $\sigma$-finite, nonatomic, Baire measure space. Let $G$ be a finite dimensional subspace of $L_1(X,\Sigma, \mu)$. There is a bounded, continuous function, $q$, defined on $X$, such that

(1) $\int _X g\operatorname{sgn} q d\mu =0$ for all $g \in G$, and (2) $|\operatorname{sgn} q | =1$ almost everywhere.

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Additional Information:

Daniel Wulbert
Affiliation: Mathematics Department 0112, University of California, San Diego, La Jolla, California 92093
Email: dwulbert@ucsd.edu

DOI: 10.1090/S0002-9939-99-05317-4
PII: S 0002-9939(99)05317-4
Keywords: Nonatomic Baire measures, Phelps-Dye theorem, Gohberg-Krein theorem, $l_1(X, \Sigma, \mu)$, extreme annihilators, continuous functions with level sets of measure zero
Received by editor(s): May 28, 1998
Received by editor(s) in revised form: September 25, 1998
Posted: November 24, 1999
Communicated by: Dale Alspach
Copyright of article: Copyright 2000, American Mathematical Society

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