On the extensions of vectorvalued Loeb measures
Authors:
Horst Osswald and Yeneng Sun
Journal:
Proc. Amer. Math. Soc. 111 (1991), 663675
MSC:
Primary 28E05; Secondary 03H05, 28B05, 46G10, 46S20
MathSciNet review:
1047007
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Abstract: Two ways of constructing countably additive vector measures from internal vector measures are given. The connection of the extendability of vectorvalued Loeb measures and the existence of the internal control measures is shown.
 [1]
J.
Diestel and J.
J. Uhl Jr., Vector measures, American Mathematical Society,
Providence, R.I., 1977. With a foreword by B. J. Pettis; Mathematical
Surveys, No. 15. MR 0453964
(56 #12216)
 [2]
A.
Grothendieck, Sur les applications linéaires faiblement
compactes d’espaces du type 𝐶(𝐾), Canadian J.
Math. 5 (1953), 129–173 (French). MR 0058866
(15,438b)
 [3]
C.
Ward Henson and L.
C. Moore Jr., Nonstandard analysis and the theory of Banach
spaces, Nonstandard analysis—recent developments (Victoria,
B.C., 1980) Lecture Notes in Math., vol. 983, Springer, Berlin,
1983, pp. 27–112. MR 698954
(85f:46033), http://dx.doi.org/10.1007/BFb0065334
 [4]
Albert
E. Hurd and Peter
A. Loeb, An introduction to nonstandard real analysis, Pure
and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL,
1985. MR
806135 (87d:03184)
 [5]
Peter
A. Loeb, Conversion from nonstandard to
standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 0390154
(52 #10980), http://dx.doi.org/10.1090/S00029947197503901548
 [6]
Peter
A. Loeb, A functional approach to nonstandard measure theory,
Conference in modern analysis and probability (New Haven, Conn., 1982),
Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984,
pp. 251–261. MR 737406
(86b:28026), http://dx.doi.org/10.1090/conm/026/737406
 [7]
P. A. Loeb and H. Osswald, Nonstandard integration theory in solid Riesz space (to appear).
 [8]
Yeneng
Sun, On the theory of vector valued Loeb measures and
integration, J. Funct. Anal. 104 (1992), no. 2,
327–362. MR 1153991
(93a:46146), http://dx.doi.org/10.1016/00221236(92)900043
 [9]
Yeneng
Sun, A nonstandard proof of the Riesz representation theorem for
weakly compact operators on 𝐶(Ω), Math. Proc. Cambridge
Philos. Soc. 105 (1989), no. 1, 141–145. MR 966150
(89i:47059), http://dx.doi.org/10.1017/S0305004100001468
 [10]
Yeneng
Sun, A Banach space in which a ball is contained in the range of
some countably additive measure is superreflexive, Canad. Math. Bull.
33 (1990), no. 1, 45–49. MR 1036854
(91a:46018), http://dx.doi.org/10.4153/CMB19900075
 [11]
, Nonstandard theory of vector measures, Ph.D. dissertation, University of Illinois, Urbana, Illinois, 1989.
 [12]
Rade
T. Živaljević, Loeb completion of internal
vectorvalued measures, Math. Scand. 56 (1985),
no. 2, 276–286. MR 813641
(87k:28021)
 [1]
 J. Diestel and J. J. Uhl, Vector measures, Mathematical Surveys Number 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
 [2]
 A. Grothendieck, Sur les applications linearies faiblement compactes d'espaces du type , Canad. J. Math. 5 (1953), 129173. MR 0058866 (15:438b)
 [3]
 C. W. Henson and L. C. Moore, Nonstandard analysis and theory of Banach spaces, Nonstandard analysis recent developments (A. E. Hurd, ed.), Lecture Notes in Math., vol. 983, SpringerVerlag, Berlin, 1983. MR 698954 (85f:46033)
 [4]
 A. E. Hurd and P. A. Loeb, An introduction to nonstandard real analysis, Academic Press, Orlando, Florida, 1985. MR 806135 (87d:03184)
 [5]
 P. A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113122. MR 0390154 (52:10980)
 [6]
 , A functional approach to nonstandard measure theory, Contemp. Math. 26 (1984), 251261. MR 737406 (86b:28026)
 [7]
 P. A. Loeb and H. Osswald, Nonstandard integration theory in solid Riesz space (to appear).
 [8]
 Y. Sun, On the theory of vector valued Loeb measures and integration (to appear). MR 1153991 (93a:46146)
 [9]
 , A nonstandard proof of the Riesz representation theorem for weakly compact operators on , Math. Proc. Comb. Phil. Soc. 105 (1989), 141145. MR 966150 (89i:47059)
 [10]
 , A Banach space in which a ball is contained in the range of some countably additive measure is superreflexive, Canad. Math. Bull. 33 (1990), 4549. MR 1036854 (91a:46018)
 [11]
 , Nonstandard theory of vector measures, Ph.D. dissertation, University of Illinois, Urbana, Illinois, 1989.
 [12]
 R. T. Zivaljevic, Loeb completion of internal vector valued measures, Math. Scand. 56 (1985), 276286. MR 813641 (87k:28021)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110470076
PII:
S 00029939(1991)10470076
Article copyright:
© Copyright 1991
American Mathematical Society
