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 & Continuous Dynamical Systems - A, 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, (1956). Google Scholar
[2]	N. Dunford and J. T. Schwartz, "Linear Operators, Part I: General Theory,", Wiley-Interscience, (1958). Google Scholar
[3]	P. L. Duren, "Theory of $H^p$ Spaces,", Academic Press, (1970). Google Scholar
[4]	W. Feller, "An Introduction to Probability Theory and its Applications, Vol. 2,", Wiley, (1966). Google Scholar
[5]	D. J. H. Garling, Non-negative random measures and order preserving embeddings,, J. London Math. Soc. (2), 11 (1975), 35. 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. 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. 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. doi: 10.1090/S0002-9939-1952-0047250-4. Google Scholar
[9]	T. V. Panchapagesan, "The Bartle-Dunford-Schwartz Integral,", Birkhäuser Verlag AG, (2008). Google Scholar
[10]	A. Prékopa, On stochastic set functions, I-III,, Acta Math. Acad. Sci. Hungary, 8 (1956), 215. doi: 10.1007/BF02020323. Google Scholar
[11]	M. M. Rao, Random measures and applications,, Stochastic Anal. Appl., 27 (2009), 1014. doi: 10.1080/07362990903136546. Google Scholar
[12]	M. M. Rao, "Random and Vector Measures,", World Scientific, (2012). Google Scholar
[13]	M. M. Rao, "Measure Theory and Integration,", Wiley-Interscience, (1987). Google Scholar
[14]	M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces,", Marcel Dekker, (1991). Google Scholar
[15]	M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces,", Marcel Dekker, (2002). doi: 10.1201/9780203910863. Google Scholar
[16]	S. Rolewicz, "Metric Linear Spaces,", Warsaw, (1972). Google Scholar
[17]	I. Shragin, "Superpositional Measurability and Superposition Operator, (Selected Themes),", Odessa, (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. 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. 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. Google Scholar
[21]	V. M. Zolotarev, "One Dimensional Stable Distributions,", Translatios A.M.S., 65 (1986). Google Scholar

show all references

References:

[1]	S. Bochner, "Harmonic Analysis and the Theory of Probability,", University of California Press, (1956). Google Scholar
[2]	N. Dunford and J. T. Schwartz, "Linear Operators, Part I: General Theory,", Wiley-Interscience, (1958). Google Scholar
[3]	P. L. Duren, "Theory of $H^p$ Spaces,", Academic Press, (1970). Google Scholar
[4]	W. Feller, "An Introduction to Probability Theory and its Applications, Vol. 2,", Wiley, (1966). Google Scholar
[5]	D. J. H. Garling, Non-negative random measures and order preserving embeddings,, J. London Math. Soc. (2), 11 (1975), 35. 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. 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. 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. doi: 10.1090/S0002-9939-1952-0047250-4. Google Scholar
[9]	T. V. Panchapagesan, "The Bartle-Dunford-Schwartz Integral,", Birkhäuser Verlag AG, (2008). Google Scholar
[10]	A. Prékopa, On stochastic set functions, I-III,, Acta Math. Acad. Sci. Hungary, 8 (1956), 215. doi: 10.1007/BF02020323. Google Scholar
[11]	M. M. Rao, Random measures and applications,, Stochastic Anal. Appl., 27 (2009), 1014. doi: 10.1080/07362990903136546. Google Scholar
[12]	M. M. Rao, "Random and Vector Measures,", World Scientific, (2012). Google Scholar
[13]	M. M. Rao, "Measure Theory and Integration,", Wiley-Interscience, (1987). Google Scholar
[14]	M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces,", Marcel Dekker, (1991). Google Scholar
[15]	M. M. Rao and Z. D. Ren, "Applications of Orlicz Spaces,", Marcel Dekker, (2002). doi: 10.1201/9780203910863. Google Scholar
[16]	S. Rolewicz, "Metric Linear Spaces,", Warsaw, (1972). Google Scholar
[17]	I. Shragin, "Superpositional Measurability and Superposition Operator, (Selected Themes),", Odessa, (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. 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. 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. Google Scholar
[21]	V. M. Zolotarev, "One Dimensional Stable Distributions,", Translatios A.M.S., 65 (1986). Google Scholar

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