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

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.
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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