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November  2013, 33(11&12): 5429-5440. doi: 10.3934/dcds.2013.33.5429

Integration with vector valued measures

M. M. Rao 1,

1. 

Unversity of California, Riverside, Riverside, CA 92521, Uruguay

Received  August 2011 Published  May 2013

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.
Keywords: independently valued measures, Vector measures in Fréchet spaces, prediction theory., stochastic measures in generalized Orlicz spaces, controling measures.
Mathematics Subject Classification: Primary: 46G10, 60H05; Secondary: 28B05, 28C2.
Citation: M. M. Rao. Integration with vector valued measures. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5429-5440. doi: 10.3934/dcds.2013.33.5429
References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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.  Google Scholar

[19]

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

[20]

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

[21]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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.  Google Scholar

[6]

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

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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.  Google Scholar

[19]

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

[20]

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

[21]

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

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