Integration with vector valued measures

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

Integration with vector valued measures

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

Received August 2011 Published May 2013

Abstract
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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 and 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.
[2]	N. Dunford and J. T. Schwartz , "Linear Operators, Part I: General Theory," Wiley-Interscience, New York, 1958.
[3]	P. L. Duren, "Theory of $H^p$ Spaces," Academic Press, New York, 1970.
[4]	W. Feller, "An Introduction to Probability Theory and its Applications, Vol. 2," Wiley, New York, 1966. .
[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.
[6]	S. Kakutani, Über die Metrisation der topologischen Grouppen, Proc. Imp. Acad. Tokyo, 12,(1936), 82-84. doi: 10.3792/pia/1195580206.
[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.
[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.
[9]	T. V. Panchapagesan, "The Bartle-Dunford-Schwartz Integral," Birkhäuser Verlag AG, Basel, (2008).
[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.
[11]	M. M. Rao, Random measures and applications, Stochastic Anal. Appl., 27, (2009), 1014-1076. doi: 10.1080/07362990903136546.
[12]	M. M. Rao, "Random and Vector Measures," World Scientific, Singapore, 2012.
[13]	M. M. Rao, "Measure Theory and Integration," Wiley-Interscience, and Marcel Dekker, New York, 1987, 2nd ed., 2004.
[14]	M. M. Rao and Z. D. Ren , "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.
[15]	M. M. Rao and Z. D. Ren , "Applications of Orlicz Spaces," Marcel Dekker, New York, 2002. doi: 10.1201/9780203910863.
[16]	S. Rolewicz, "Metric Linear Spaces," Warsaw, Poland, 1972.
[17]	I. Shragin, "Superpositional Measurability and Superposition Operator, (Selected Themes)," Odessa, "Astroprint'', 2007.
[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.
[19]	M. Talagrand, Les mesures vectorielles a valuers dans $L^0$ sont bournées, Ann. Sci. Ècole Norm. asup., 14,(1981), 445-452.
[20]	K. Urbanik, Some prediction problems for strictly stationary processes, Proc. 5th Berkely Symp. Math. Statist. and Prob., 2, part 1, (1967), 235-258.
[21]	V. M. Zolotarev, "One Dimensional Stable Distributions," Translatios A.M.S., 65, Providence, R.I., 1986.

show all references

References:

[1]	S. Bochner, "Harmonic Analysis and the Theory of Probability," University of California Press, Berkely, CA, 1956.
[2]	N. Dunford and J. T. Schwartz , "Linear Operators, Part I: General Theory," Wiley-Interscience, New York, 1958.
[3]	P. L. Duren, "Theory of $H^p$ Spaces," Academic Press, New York, 1970.
[4]	W. Feller, "An Introduction to Probability Theory and its Applications, Vol. 2," Wiley, New York, 1966. .
[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.
[6]	S. Kakutani, Über die Metrisation der topologischen Grouppen, Proc. Imp. Acad. Tokyo, 12,(1936), 82-84. doi: 10.3792/pia/1195580206.
[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.
[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.
[9]	T. V. Panchapagesan, "The Bartle-Dunford-Schwartz Integral," Birkhäuser Verlag AG, Basel, (2008).
[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.
[11]	M. M. Rao, Random measures and applications, Stochastic Anal. Appl., 27, (2009), 1014-1076. doi: 10.1080/07362990903136546.
[12]	M. M. Rao, "Random and Vector Measures," World Scientific, Singapore, 2012.
[13]	M. M. Rao, "Measure Theory and Integration," Wiley-Interscience, and Marcel Dekker, New York, 1987, 2nd ed., 2004.
[14]	M. M. Rao and Z. D. Ren , "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991.
[15]	M. M. Rao and Z. D. Ren , "Applications of Orlicz Spaces," Marcel Dekker, New York, 2002. doi: 10.1201/9780203910863.
[16]	S. Rolewicz, "Metric Linear Spaces," Warsaw, Poland, 1972.
[17]	I. Shragin, "Superpositional Measurability and Superposition Operator, (Selected Themes)," Odessa, "Astroprint'', 2007.
[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.
[19]	M. Talagrand, Les mesures vectorielles a valuers dans $L^0$ sont bournées, Ann. Sci. Ècole Norm. asup., 14,(1981), 445-452.
[20]	K. Urbanik, Some prediction problems for strictly stationary processes, Proc. 5th Berkely Symp. Math. Statist. and Prob., 2, part 1, (1967), 235-258.
[21]	V. M. Zolotarev, "One Dimensional Stable Distributions," Translatios A.M.S., 65, Providence, R.I., 1986.

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