Approximation of Jensen measures by image measures under holomorphic functions and applications

Bu, Shang Quan; Schachermayer, Walter

doi:10.1090/S0002-9947-1992-1035999-6

Approximation of Jensen measures by image measures under holomorphic functions and applications
HTML articles powered by AMS MathViewer

by Shang Quan Bu and Walter Schachermayer

Trans. Amer. Math. Soc. 331 (1992), 585-608

DOI: https://doi.org/10.1090/S0002-9947-1992-1035999-6

PDF | Request permission

Abstract:

We show that Jensen measures defined on ${\mathbb {C}^n}$ or more generally on a complex Banach space $X$ can be approximated by the image of Lebesgue measure on the torus under $X$-valued polynomials defined on $\mathbb {C}$. We give similar characterizations for Jensen measures in terms of analytic martingales and Hardy martingales. The results are applied to approximate plurisubharmonic martingales by Hardy martingales, which enables us to give a characterization of the analytic Radon-Nikodym property of Banach spaces in terms of convergence of plurisubharmonic martingales, thus solving a problem of G. A. Edgar.

References

J. Bourgain and W. J. Davis, Martingale transforms and complex uniform convexity, Trans. Amer. Math. Soc. 294 (1986), no. 2, 501–515. MR 825718, DOI 10.1090/S0002-9947-1986-0825718-5
A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in a Banach space, Mat. Zametki 31 (1982), no. 2, 203–214, 317 (Russian). MR 649004
Soo Bong Chae, Holomorphy and calculus in normed spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 92, Marcel Dekker, Inc., New York, 1985. With an appendix by Angus E. Taylor. MR 788158
Donald L. Cohn, Measure theory, Birkhäuser, Boston, Mass., 1980. MR 578344
William J. Davis, D. J. H. Garling, and Nicole Tomczak-Jaegermann, The complex convexity of quasinormed linear spaces, J. Funct. Anal. 55 (1984), no. 1, 110–150. MR 733036, DOI 10.1016/0022-1236(84)90021-1
Patrick N. Dowling, Representable operators and the analytic Radon-Nikodým property in Banach spaces, Proc. Roy. Irish Acad. Sect. A 85 (1985), no. 2, 143–150. MR 845538
Richard Durrett, Brownian motion and martingales in analysis, Wadsworth Mathematics Series, Wadsworth International Group, Belmont, CA, 1984. MR 750829
G. A. Edgar, Complex martingale convergence, Banach spaces (Columbia, Mo., 1984) Lecture Notes in Math., vol. 1166, Springer, Berlin, 1985, pp. 38–59. MR 827757, DOI 10.1007/BFb0074691
G. A. Edgar, Analytic martingale convergence, J. Funct. Anal. 69 (1986), no. 2, 268–280. MR 865224, DOI 10.1016/0022-1236(86)90092-3
G. A. Edgar, Extremal integral representations, J. Functional Analysis 23 (1976), no. 2, 145–161. MR 0435797, DOI 10.1016/0022-1236(76)90072-0
T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. MR 521440
D. J. H. Garling, On martingales with values in a complex Banach space, Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 2, 399–406. MR 948923, DOI 10.1017/S0305004100065555
N. Ghoussoub and B. Maurey, $G_\delta$-embeddings in Hilbert space. II, J. Funct. Anal. 78 (1988), no. 2, 271–305. MR 943500, DOI 10.1016/0022-1236(88)90121-8
J. Hoffmann-Jørgensen, The theory of analytic spaces, Various Publications Series, No. 10, Aarhus Universitet, Matematisk Institut, Aarhus, 1970. MR 0409748
Steven G. Krantz, Function theory of several complex variables, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1982. MR 635928
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056
R. Michael Range, Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, vol. 108, Springer-Verlag, New York, 1986. MR 847923, DOI 10.1007/978-1-4757-1918-5

Disintegration of measures

Laurent Schwartz, Radon measures on arbitrary topological spaces and cylindrical measures, Tata Institute of Fundamental Research Studies in Mathematics, No. 6, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1973. MR 0426084