American Institute of Mathematical Sciences

Advanced
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 - 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

[1]

Barry Simon. Equilibrium measures and capacities in spectral theory. Inverse Problems & Imaging, 2007, 1 (4) : 713-772. doi: 10.3934/ipi.2007.1.713
[2]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[3]

Zeng Lian, Peidong Liu, Kening Lu. Existence of SRB measures for a class of partially hyperbolic attractors in banach spaces. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3905-3920. doi: 10.3934/dcds.2017164
[4]

Moisey Guysinsky, Serge Yaskolko. Coincidence of various dimensions associated with metrics and measures on metric spaces. Discrete & Continuous Dynamical Systems - A, 1997, 3 (4) : 591-603. doi: 10.3934/dcds.1997.3.591
[5]

J. Alberto Conejero, Marko Kostić, Pedro J. Miana, Marina Murillo-Arcila. Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1915-1939. doi: 10.3934/cpaa.2016022
[6]

G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1
[7]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[8]

Vasso Anagnostopoulou. Stochastic dominance for shift-invariant measures. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 667-682. doi: 10.3934/dcds.2019027
[9]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[10]

Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511
[11]

Xu Zhang. Sinai-Ruelle-Bowen measures for piecewise hyperbolic maps with two directions of instability in three-dimensional spaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2873-2886. doi: 10.3934/dcds.2016.36.2873
[12]

Xin Li, Wenxian Shen, Chunyou Sun. Invariant measures for complex-valued dissipative dynamical systems and applications. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2427-2446. doi: 10.3934/dcdsb.2017124
[13]

Siegfried Carl, Christoph Tietz. Quasilinear elliptic equations with measures and multi-valued lower order terms. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 193-212. doi: 10.3934/dcdss.2018012
[14]

S. Astels. Thickness measures for Cantor sets. Electronic Research Announcements, 1999, 5: 108-111.

[15]

François Golse, Clément Mouhot, Valeria Ricci. Empirical measures and Vlasov hierarchies. Kinetic & Related Models, 2013, 6 (4) : 919-943. doi: 10.3934/krm.2013.6.919
[16]

Moshe Marcus. Remarks on nonlinear equations with measures. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1745-1753. doi: 10.3934/cpaa.2013.12.1745
[17]

Azmy S. Ackleh, Rinaldo M. Colombo, Sander C. Hille, Adrian Muntean. Preface to ``Modeling with Measures". Mathematical Biosciences & Engineering, 2015, 12 (2) : i-ii. doi: 10.3934/mbe.2015.12.2i
[18]

Alexander I. Bufetov. Infinite determinantal measures. Electronic Research Announcements, 2013, 20: 12-30. doi: 10.3934/era.2013.20.12
[19]

Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1299-1309. doi: 10.3934/dcds.2010.28.1299
[20]

Ilie Ugarcovici. On hyperbolic measures and periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 505-512. doi: 10.3934/dcds.2006.16.505

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences