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Integral representation of linear functionals on spaces of unbounded functions

Authors: Patrizia Berti and Pietro Rigo
Journal: Proc. Amer. Math. Soc. 128 (2000), 3251-3258
MSC (2000): Primary 28C05; Secondary 60A05
Published electronically: April 28, 2000
MathSciNet review: 1694449
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Let $L$ be a vector lattice of real functions on a set $\Omega$ with $\boldsymbol{1}\in L$, and let $P$ be a linear positive functional on $L$. Conditions are given which imply the representation $P(f)=\int fd\pi$, $f\in L$, for some bounded charge $\pi$. As an application, for any bounded charge $\pi$ on a field $\mathcal F$, the dual of $L^1(\pi)$ is shown to be isometrically isomorphic to a suitable space of bounded charges on $\mathcal F$. In addition, it is proved that, under one more assumption on $L$, $P$ is the integral with respect to a $\sigma$-additive bounded charge.

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Additional Information

Patrizia Berti
Affiliation: Dipartimento di Matematica Pura ed Applicata “G. Vitali”, Università di Modena, via Campi 213/B, 41100 Modena, Italy

Pietro Rigo
Affiliation: Dipartimento di Statistica “G. Parenti”, Università di Firenze, viale Morgagni 59, 50134 Firenze, Italy

Keywords: Bounded charge, expectation, integral representation, linear positive functional
Received by editor(s): August 1, 1997
Received by editor(s) in revised form: December 17, 1998
Published electronically: April 28, 2000
Additional Notes: This research was partially supported by M.U.R.S.T. 40% “Processi Stocastici”.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society