A type of Strassen’s theorem for positive vector measures with values in dual spaces

Kawabe, Jun

doi:10.1090/S0002-9939-00-05384-3

A type of Strassen’s theorem for positive vector measures with values in dual spaces
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by Jun Kawabe PDF

Proc. Amer. Math. Soc. 128 (2000), 3291-3300 Request permission

Abstract:

In this paper, we extend a type of Strassen’s theorem for the existence of probability measures with given marginals to positive vector measures with values in the dual of a barreled locally convex space which has certain order conditions. In this process of the extension we also give some useful properties for vector measures with values in dual spaces.

References

M. Dekiert, Kompaktheit, Fortsetzbarkeit und Konvergenz von Vectormaßen, Dissertation, University of Essen, 1991.
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964, DOI 10.1090/surv/015
Richard M. Dudley, Real analysis and probability, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989. MR 982264
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 Wiley-Interscience Publication. MR 1009162
D. A. Edwards, On the existence of probability measures with given marginals, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 4, 53–78, v (English, with French summary). MR 513882, DOI 10.5802/aif.717
J. Hoffmann-Jørgensen, Probability in Banach space, École d’Été de Probabilités de Saint-Flour, VI-1976, Lecture Notes in Math., Vol. 598, Springer-Verlag, Berlin, 1977, pp. 1–186. MR 0461610
A. Hirshberg and R. M. Shortt, A version of Strassen’s theorem for vector-valued measures, Proc. Amer. Math. Soc. 126 (1998), no. 6, 1669–1671. MR 1443832, DOI 10.1090/S0002-9939-98-04236-1
Hans Jarchow, Locally convex spaces, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981. MR 632257, DOI 10.1007/978-3-322-90559-8
J. L. Kelley and Isaac Namioka, Linear topological spaces, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. MR 0166578, DOI 10.1007/978-3-662-41914-4
Igor Kluvánek and Greg Knowles, Vector measures and control systems, North-Holland Mathematics Studies, Vol. 20, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. MR 0499068
Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
D. R. Lewis, Integration with respect to vector measures, Pacific J. Math. 33 (1970), 157–165. MR 259064, DOI 10.2140/pjm.1970.33.157
Michael März and R. M. Shortt, Weak convergence of vector measures, Publ. Math. Debrecen 45 (1994), no. 1-2, 71–92. MR 1291803
Charles W. McArthur, On a theorem of Orlicz and Pettis, Pacific J. Math. 22 (1967), 297–302. MR 213848, DOI 10.2140/pjm.1967.22.297
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 27, Springer-Verlag, Berlin-New York, 1970. Second printing. MR 0257800, DOI 10.1007/978-3-662-35155-0
R. M. Shortt, Strassen’s marginal problem in two or more dimensions, Z. Wahrsch. Verw. Gebiete 64 (1983), no. 3, 313–325. MR 716489, DOI 10.1007/BF00532964
Heinz J. Skala, The existence of probability measures with given marginals, Ann. Probab. 21 (1993), no. 1, 136–142. MR 1207218
O. G. Smoljanov and S. V. Fomīn, Measures on topological linear spaces, Usephi Mat. Nauk 31 (1976), no. 4, (190), 3–56 (Russian). MR 0420764, DOI 10.1070/rm1976v031n04abeh001553
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist. 36 (1965), 423–439. MR 177430, DOI 10.1214/aoms/1177700153
Flemming Topsøe, Topology and measure, Lecture Notes in Mathematics, Vol. 133, Springer-Verlag, Berlin-New York, 1970. MR 0422560, DOI 10.1007/BFb0069481
François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
V. S. Varadarajan, Measures on topological spaces, Mat. Sb. (N.S.) 55 (97) (1961), 35–100 (Russian). MR 0148838