The Carathéodory extension theorem for vector valued measures
Author:
Joseph Kupka
Journal:
Proc. Amer. Math. Soc. 72 (1978), 5761
MSC:
Primary 46G10; Secondary 28A45, 60H05
MathSciNet review:
0493327
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Abstract: This paper comprises three advertisements for a known theorem which, the author believes, deserves the title of the Carathéodory extension theorem for vector valued premeasures. Principal among these is a short and transparent proof of Porcelli's criterion for the weak convergence of a sequence in the Banach space of bounded finitely additive complex measures defined on an arbitrary field, and equipped with the total variation norm. Also, a characterization of the socalled Carathéodory Extension Property is presented, and there is a brief discussion of the relevance of this material to stochastic integration.
 [1]
R.
G. Bartle, A general bilinear vector integral, Studia Math.
15 (1956), 337–352. MR 0080721
(18,289a)
 [2]
James
K. Brooks, On the existence of a control measure
for strongly bounded vector measures, Bull.
Amer. Math. Soc. 77
(1971), 999–1001. MR 0286971
(44 #4178), http://dx.doi.org/10.1090/S000299041971128343
 [3]
James
K. Brooks, Weak compactness in the space of
vector measures, Bull. Amer. Math. Soc. 78 (1972), 284–287.
MR
0324408 (48 #2760), http://dx.doi.org/10.1090/S000299041972129604
 [4]
James
K. Brooks and Robert
S. Jewett, On finitely additive vector measures, Proc. Nat.
Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 0269802
(42 #4697)
 [5]
R.
B. Darst, A direct proof of Porcelli’s
condition for weak convergence, Proc. Amer.
Math. Soc. 17
(1966), 1094–1096. MR 0206687
(34 #6505), http://dx.doi.org/10.1090/S00029939196602066870
 [6]
J.
Diestel, Applications of weak compactness and bases to vector
measures and vectorial integration, Rev. Roumaine Math. Pures Appl.
18 (1973), 211–224. MR 0317042
(47 #5590)
 [7]
Nelson
Dunford and Jacob
T. Schwartz, Linear Operators. I. General Theory, With the
assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics,
Vol. 7, Interscience Publishers, Inc., New York, 1958. MR 0117523
(22 #8302)
 [8]
Stelian
Găină, Extension of vector measures, Rev. Math.
Pures Appl. 8 (1963), 151–154. MR 0163998
(29 #1297)
 [9]
Igor
Kluvánek, The extension and closure of vector measure,
Vector and operator valued measures and applications (Proc. Sympos., Alta,
Utah, 1972), Academic Press, New York, 1973, pp. 175–190. MR 0335741
(49 #521)
 [10]
A.
U. Kussmaul, Stochastic integration and generalized
martingales, Pitman Publishing, LondonSan Francisco,
Calif.Melbourne, 1977. Research Notes in Mathematics, No. 11. MR 0488281
(58 #7841)
 [11]
Michel
Métivier, Stochastic integral and vector valued
measures, Vector and operator valued measures and applications (Proc.
Sympos., Alta, Utah, 1972), Academic Press, New York, 1973,
pp. 283–296. MR 0331509
(48 #9842)
 [12]
, The stochastic integral with respect to processes with values in a reflexive Banach space, Theor. Probability Appl. 19 (1974), 758787.
 [13]
J. Pellaumail, Thèse, Rennes, France, 1972.
 [14]
R.
S. Phillips, On weakly compact subsets of a Banach space,
Amer. J. Math. 65 (1943), 108–136. MR 0007938
(4,218f)
 [15]
Pasquale
Porcelli, Two embedding theorems with applications to weak
convergence and compactness in spaces of additive type functions, J.
Math. Mech. 9 (1960), 273–292. MR 0124723
(23 #A2034)
 [16]
Maurice
Sion, A theory of semigroup valued measures, Lecture Notes in
Mathematics, Vol. 355, SpringerVerlag, Berlin, 1973. MR 0450503
(56 #8797)
 [17]
Maurice
Sion, Outer measures with values in a topological group, Proc.
London Math. Soc. (3) 19 (1969), 89–106. MR 0239039
(39 #398)
 [18]
T.
Traynor, 𝑆bounded additive set functions, Vector and
operator valued measures and applications (Proc. Sympos., Alta, Utah,
1972), Academic Press, New York, 1973, pp. 355–365. MR 0333115
(48 #11440)
 [1]
 R. C. Bartle, A general bilinear vector integral, Studia Math. 15 (1956), 337352. MR 0080721 (18:289a)
 [2]
 J. K. Brooks, On the existence of a control measure for strongly bounded vector measures, Bull. Amer. Math. Soc. 77 (1971), 9991001. MR 0286971 (44:4178)
 [3]
 , Weak compactness in the space of vector measures, Bull. Amer. Math. Soc. 78 (1972), 284287. MR 0324408 (48:2760)
 [4]
 J. K. Brooks and R. S. Jewitt, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 12941298. MR 0269802 (42:4697)
 [5]
 R. B. Darst, A direct proof of Porcelli's condition for weak convergence, Proc. Amer. Math. Soc. 17 (1966), 10941096. MR 0206687 (34:6505)
 [6]
 J. Diestel, Applications of weak compactness and bases to vectorial integration, Rev. Roumaine Math. Pures et Appl. 18 (1973), 211224. MR 0317042 (47:5590)
 [7]
 N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR 0117523 (22:8302)
 [8]
 S. Gaina, Extension of vector measures, Rev. Roumaine Math. Pures et Appl. 8 (1963), 151154. MR 0163998 (29:1297)
 [9]
 I. Kluvánek, The extension and closure of vector measure, Vector and Operator Valued Measures and Applications (Proc. Sympos., Alta, Utah, 1972), Academic Press, New York, 1973, pp. 175190. MR 0335741 (49:521)
 [10]
 A. U. Kussmaul, Stochastic integration and generalized martingales, Research Notes in Math., vol. 11, Pitman, London, 1977. MR 0488281 (58:7841)
 [11]
 M. Metivier, Stochastic integral and vector valued measures, Vector and Operator Valued Measures and Applications (Proc. Sympos., Alta, Utah, 1972), Academic Press, New York, 1973, pp. 283296. MR 0331509 (48:9842)
 [12]
 , The stochastic integral with respect to processes with values in a reflexive Banach space, Theor. Probability Appl. 19 (1974), 758787.
 [13]
 J. Pellaumail, Thèse, Rennes, France, 1972.
 [14]
 R. S. Phillips, On weakly compact subsets of a Banach space, Amer. J. Math. 65 (1943), 108136. MR 0007938 (4:218f)
 [15]
 P. Porcelli, Two embedding theorems with applications to weak convergence and compactness in spaces of additive type functions, J. Math. Mech. 9 (1960), 273292. MR 0124723 (23:A2034)
 [16]
 M. Sion, A theory of semigroup valued measures, Lecture Notes in Math., vol. 355, SpringerVerlag, Berlin and New York, 1973. MR 0450503 (56:8797)
 [17]
 , Outer measures with values in a topological group, Proc. London Math. Soc. 19 (1969), 89106. MR 0239039 (39:398)
 [18]
 T. Traynor, Sbounded additive set functions, Vector and Operator Valued Measures and Applications (Proc. Sympos., Alta, Utah, 1972), Academic Press, New York, 1973, pp. 355365. MR 0333115 (48:11440)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197804933277
PII:
S 00029939(1978)04933277
Keywords:
Vector valued premeasures,
weak convergence of measures,
Carathéodory Extension Property,
stochastic integration
Article copyright:
© Copyright 1978 American Mathematical Society
