Advanced Search
Article Contents
Article Contents

Integration with vector valued measures

Abstract Related Papers Cited by
  • Of the many variations of vector measures, the Fréchet variation is finite valued but only subadditive. Finding a `controlling' finite measure for these in several cases, it is possible to develop a useful integration of the Bartle-Dunford-Schwartz type for many linear metric spaces. These include the generalized Orlicz spaces, $L^{\varphi}(\mu)$, where $\varphi$ is a concave $\varphi$-function with applications to stochastic measures $Z(\cdot)$ into various Fréchet spaces useful in prediction theory. In particular, certain $p$-stable random measures and a (sub) class of these leading to positive infinitely divisible ones are detailed.
    Mathematics Subject Classification: Primary: 46G10, 60H05; Secondary: 28B05, 28C20.


    \begin{equation} \\ \end{equation}
  • [1]

    S. Bochner, "Harmonic Analysis and the Theory of Probability," University of California Press, Berkely, CA, 1956.


    N. Dunford and J. T. Schwartz , "Linear Operators, Part I: General Theory," Wiley-Interscience, New York, 1958.


    P. L. Duren, "Theory of $H^p$ Spaces," Academic Press, New York, 1970.


    W. Feller, "An Introduction to Probability Theory and its Applications, Vol. 2," Wiley, New York, 1966. .


    D. J. H. Garling, Non-negative random measures and order preserving embeddings, J. London Math. Soc. (2), 11, (1975), 35-45. .doi: 10.1112/jlms/s2-11.1.35.


    S. Kakutani, Über die Metrisation der topologischen Grouppen, Proc. Imp. Acad. Tokyo, 12,(1936), 82-84.doi: 10.3792/pia/1195580206.


    N. J. Kalton, N. T. Peck and J. W. Roberts, $L^0$-valued vector measures are bounded, Proc. Amer. Math. Soc., 85, (1982), 575-582.doi: 10.2307/2044069.


    V. L. Klee, Invariant metrics in groups:(Solution of a problem of Banach), Proc. Amer. Math. Soc., 3, (1952), 484-487.doi: 10.1090/S0002-9939-1952-0047250-4.


    T. V. Panchapagesan, "The Bartle-Dunford-Schwartz Integral," Birkhäuser Verlag AG, Basel, (2008).


    A. Prékopa, On stochastic set functions, I-III, Acta Math. Acad. Sci. Hungary, 8, (1956), 215-263; (1957),337-374; 375-400.doi: 10.1007/BF02020323.


    M. M. Rao, Random measures and applications, Stochastic Anal. Appl., 27, (2009), 1014-1076.doi: 10.1080/07362990903136546.


    M. M. Rao, "Random and Vector Measures," World Scientific, Singapore, 2012.


    M. M. Rao, "Measure Theory and Integration," Wiley-Interscience, and Marcel Dekker, New York, 1987, 2nd ed., 2004.


    M. M. Rao and Z. D. Ren , "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.


    M. M. Rao and Z. D. Ren , "Applications of Orlicz Spaces," Marcel Dekker, New York, 2002.doi: 10.1201/9780203910863.


    S. Rolewicz, "Metric Linear Spaces," Warsaw, Poland, 1972.


    I. Shragin, "Superpositional Measurability and Superposition Operator, (Selected Themes)," Odessa, "Astroprint'', 2007.


    M. S. Steigerwalt and A. J. White , Some function spaces related to $L_p$, Proc. London Math. Soc., 22, (1971), 137-163.doi: 10.1112/plms/s3-22.1.137.


    M. Talagrand, Les mesures vectorielles a valuers dans $L^0$ sont bournées, Ann. Sci. Ècole Norm. asup., 14,(1981), 445-452.


    K. Urbanik, Some prediction problems for strictly stationary processes, Proc. 5th Berkely Symp. Math. Statist. and Prob., 2, part 1, (1967), 235-258.


    V. M. Zolotarev, "One Dimensional Stable Distributions," Translatios A.M.S., 65, Providence, R.I., 1986.

  • 加载中

Article Metrics

HTML views() PDF downloads(212) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint