Integral representation of linear functionals on spaces of unbounded functions
Authors:
Patrizia Berti and Pietro Rigo
Journal:
Proc. Amer. Math. Soc. 128 (2000), 32513258
MSC (2000):
Primary 28C05; Secondary 60A05
Published electronically:
April 28, 2000
MathSciNet review:
1694449
Fulltext PDF Free Access
Abstract 
References 
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Abstract: 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.
Patrizia
Berti, Eugenio
Regazzini, and Pietro
Rigo, Finitely additive RadonNikodým
theorem and concentration function of a probability with respect to a
probability, Proc. Amer. Math. Soc.
114 (1992), no. 4,
1069–1078. MR 1045586
(92g:60006), http://dx.doi.org/10.1090/S00029939199210455867
 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.
Stephen
T. L. Choy and James
C. S. Wong, A characterization of the second dual
of 𝐶₀(𝑆,𝐴), Proc. Amer. Math. Soc. 120 (1994), no. 1, 203–211. MR 1163330
(94b:46057), http://dx.doi.org/10.1090/S00029939199411633301
 5.
Lester
E. Dubins, On everywheredefined
integrals, Trans. Amer. Math. Soc. 232 (1977), 187–194. MR 0450489
(56 #8783), http://dx.doi.org/10.1090/S00029947197704504899
 6.
Nelson
Dunford and Jacob
T. Schwartz, Linear operators. Part I, Wiley Classics Library,
John Wiley & Sons, Inc., New York, 1988. General theory; With the
assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958
original; A WileyInterscience Publication. MR 1009162
(90g:47001a)
 7.
Edwin
Hewitt, Linear functionals on spaces of continuous functions,
Fund. Math. 37 (1950), 161–189. MR 0042684
(13,147g)
 8.
Heinz
König, The DaniellStoneRiesz representation theorem,
Operator theory in function spaces and Banach lattices, Oper. Theory Adv.
Appl., vol. 75, Birkhäuser, Basel, 1995, pp. 191–222.
MR
1322505 (96f:28016)
 9.
L.
H. Loomis, Linear functionals and content, Amer. J. Math.
76 (1954), 168–182. MR 0060145
(15,631d)
 10.
Marian
Nowak, Singular linear functionals on nonlocally convex Orlicz
spaces, Indag. Math. (N.S.) 3 (1992), no. 3,
337–351. MR 1186742
(94a:46034), http://dx.doi.org/10.1016/00193577(92)90040R
 11.
V. K. Zakharov and A. V. Mikhalev (1997), Radon problem for regular measures on an arbitrary Hausdorff space, Fundam. Prikl. Mat. 3, 801808 (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, 337339. (Translated from Doklady Akademii Nauk 1998, 360, 1315). CMP 99:09
 1.
 P. Berti, E. Regazzini, and P. Rigo (1992), Finitely additive RadonNikodym theorem and concentration function of a probability with respect to a probability, Proc. Amer. Math. Soc. 114, 10691078. 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, 203211. MR 94b:46057
 5.
 L. E. Dubins (1977), On everywheredefined integrals, Trans. Amer. Math. Soc. 232, 187194. 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, 161189. MR 13:147g
 8.
 H. J. E. Konig (1995), The DaniellStoneRiesz representation theorem: Operator theory in function spaces and Banach lattices, Operator Theory: Adv. Appl., Birkhauser 75, 191222. MR 96f:28016
 9.
 L. H. Loomis (1954), Linear functionals and content, Amer. J. Math. 76, 168182. MR 15:631d
 10.
 M. Nowak (1992), Singular linear functionals on nonlocally convex Orlicz spaces, Indag. Mathem., N.S. 3 (3), 337351. 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, 801808 (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, 337339. (Translated from Doklady Akademii Nauk 1998, 360, 1315). 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:
http://dx.doi.org/10.1090/S0002993900055106
PII:
S 00029939(00)055106
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
