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

DOI:
https://doi.org/10.1090/S0002-9939-00-05510-6

Published electronically:
April 28, 2000

MathSciNet review:
1694449

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be a vector lattice of real functions on a set with , and let be a linear positive functional on . Conditions are given which imply the representation , , for some bounded charge . As an application, for any bounded charge on a field , the dual of is shown to be isometrically isomorphic to a suitable space of bounded charges on . In addition, it is proved that, under one more assumption on , is the integral with respect to a -additive bounded charge.

**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*, 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

Email:
berti.patrizia@unimo.it

**Pietro Rigo**

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

Email:
rigo@ds.unifi.it

DOI:
https://doi.org/10.1090/S0002-9939-00-05510-6

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