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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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Abstract | References | Similar Articles | Additional Information


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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  • 1. P. Berti, E. Regazzini, and P. Rigo (1992), Finitely additive Radon-Nikodym theorem and concentration function of a probability with respect to a probability, Proc. Amer. Math. Soc. 114, 1069-1078. MR 92g:60006
  • 2. P. Berti, E. Regazzini, and P. Rigo (1994), Coherent prevision of random elements, Quaderno I.A.M.I. 94.15, Milano.
  • 3. K. P. S. Bhaskara Rao and M. Bhaskara Rao (1983), Theory of charges, Academic Press, London.
  • 4. S. T. L. Choy and J. C. S. Wong (1994), A characterization of the second dual of $C_0(S,A)$, Proc. Amer. Math. Soc. 120, 203-211. MR 94b:46057
  • 5. L. E. Dubins (1977), On everywhere-defined integrals, Trans. Amer. Math. Soc. 232, 187-194. MR 56:8783
  • 6. N. Dunford and J.T. Schwartz (1958), Linear operators, Part I, General theory, Wiley, Interscience, New York. MR 90g:47001a
  • 7. E. Hewitt (1950), Linear functionals on spaces of continuous functions, Fund. Math. 37, 161-189. MR 13:147g
  • 8. H. J. E. Konig (1995), The Daniell-Stone-Riesz representation theorem: Operator theory in function spaces and Banach lattices, Operator Theory: Adv. Appl., Birkhauser 75, 191-222. MR 96f:28016
  • 9. L. H. Loomis (1954), Linear functionals and content, Amer. J. Math. 76, 168-182. MR 15:631d
  • 10. M. Nowak (1992), Singular linear functionals on non-locally convex Orlicz spaces, Indag. Mathem., N.S. 3 (3), 337-351. MR 94a:46034
  • 11. V. K. Zakharov and A. V. Mikhalev (1997), Radon problem for regular measures on an arbitrary Hausdorff space, Fundam. Prikl. Mat. 3, 801-808 (in Russian).
  • 12. V. K. Zakharov and A. V. Mikhalev (1998), The problem of integral representation for Radon measures on a Hausdorff space, Doklady Mathematics 57, 337-339. (Translated from Doklady Akademii Nauk 1998, 360, 13-15). CMP 99:09

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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

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