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Measures and Integrals in Conditional Set Theory

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Abstract

The aim of this article is to establish basic results in a conditional measure theory. The results are applied to prove that arbitrary kernels and conditional distributions are represented by measures in a conditional set theory. In particular, this extends the usual representation results for separable spaces.

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References

  1. Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis: a Hitchhiker’s Guide. Springer (2006)

  2. Backhoff, J., Horst, U.: Conditional analysis and a principal-agent problem. SIAM J. Financial Math. 7(1), 477–507 (2016)

    Article  MathSciNet  Google Scholar 

  3. Bielecki, T., Cialenco, I., Drapeau, S., Karliczek, M.: Dynamic assessement indices. Stochastics 88(1), 1–44 (2016)

    Article  MathSciNet  Google Scholar 

  4. Cerreia-Vioglio, S., Kupper, M., Maccheroni, F., Marinacci, M., Vogelpoth, N.: Conditional L p-spaces and the duality of modules over f -algebras. J. Math. Anal. Appl. 444(2), 1045–1070 (2016)

    Article  MathSciNet  Google Scholar 

  5. Cheridito, P., Hu, Y.: Optimal consumption and investment in incomplete markets with general constraints. Stoch. Dyn. 11(2), 283–299 (2011)

    Article  MathSciNet  Google Scholar 

  6. Cheridito, P., Kupper, M., Vogelpoth, N.: Conditional analysis on \(\mathbb {R}^{d}\). Set Optimization and Applications, Proceedings in Mathematics & Statistics 151, 179–211 (2015)

    Article  MathSciNet  Google Scholar 

  7. Cheridito, P., Stadje, M.: BSΔEs and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness. Bernoulli 19(3), 1047–1085 (2013)

    Article  MathSciNet  Google Scholar 

  8. Drapeau, S., Jamneshan, A.: Conditional preferences and their numerical representations. J. Math. Econom. 63, 106–118 (2016)

    Article  MathSciNet  Google Scholar 

  9. Drapeau, S., Jamneshan, A., Karliczek, M., Kupper, M.: The algebra of conditional sets, and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561–589 (2016)

    Article  MathSciNet  Google Scholar 

  10. Drapeau, S., Jamneshan, A., Kupper, M.: A Fenchel-Moreau theorem for L 0-valued functions. Preprint available at arXiv:1708.03127 (2017)

  11. Drapeau, S., Karliczek, M., Kupper, M., Streckfuss, M.: Brouwer fixed point theorem in (L 0)d. J. Fixed Point Theory Appl. 301(1), (2013). https://doi.org/10.1186/1687-1812-2013-301

    Article  Google Scholar 

  12. Filipović, D., Kupper, M., Vogelpoth, N.: Separation and duality in locally L 0-convex modules. J. Funct. Anal. 256, 3996–4029 (2009)

    Article  MathSciNet  Google Scholar 

  13. Filipović, D., Kupper, M., Vogelpoth, N.: Approaches to conditional risk. SIAM J. Financial Math. 3(1), 402–432 (2012)

    Article  MathSciNet  Google Scholar 

  14. Frittelli, M., Maggis, M.: Complete duality for quasiconvex dynamic risk measures on modules of the L p-type. Stat. Risk Model. 31(1), 103–128 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Frittelli, M., Maggis, M.: Conditionally evenly convex sets and evenly quasi-convex maps. J. Math. Anal. Appl. 413, 169–184 (2014)

    Article  MathSciNet  Google Scholar 

  16. Guo, T.: Relations between some basic results derived from two kinds of topologies for a random locally convex module. J. Funct. Anal. 258, 3024–3047 (2010)

    Article  MathSciNet  Google Scholar 

  17. Hansen, L.P., Richard, S.F.: The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models. Econometrica 55, 587–613 (1987)

    Article  MathSciNet  Google Scholar 

  18. Jamneshan, A.: Sheaves and conditional sets. Preprint available at arXiv:1403.5405 (2014)

  19. Jamneshan, A., Kupper, M., Zapata, J.M.: Parameter-dependent stochastic optimal control in finite discrete time. Preprint available at arXiv:1705.02374 (2017)

  20. Jamneshan, A., Zapata, J.M.: On compactness in L 0-modules. Preprint available at arXiv:1711.09785 (2017)

  21. Kallenberg, O.: Foundations of Modern Probability. Probability and Its Applications (New York), 2nd edn. Springer, New York (2002)

    Google Scholar 

  22. Monk, J., Koppelberg, S., Bonnet, R.: Handbook of boolean algebras. Handbook of boolean algebras: general theory of boolean algebras. North-Holland (1989)

  23. Orihuela, J., Zapata, J.M.: Stability in locally L 0-convex modules and a conditional version of James’ compactness theorem. J. Math. Anal. Appl. 452(2), 1101–1127 (2017)

    Article  MathSciNet  Google Scholar 

  24. Takeuti, G.: Two Applications of Logic to Mathematics. Publications of the Mathematical Society of Japan. Princeton University Press (2015)

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Acknowledgments

A.J. and M.K. gratefully acknowledge financial support from DFG project KU 2740/2-1. The authors would like to thank an anonymous referee for a careful reading of the manuscript and helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Konstanz, Konstanz, Germany

    Asgar Jamneshan & Michael Kupper

  2. Department of Mathematics, Humboldt University Berlin, Berlin, Germany

    Martin Streckfuß

Authors
  1. Asgar Jamneshan
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  2. Michael Kupper
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  3. Martin Streckfuß
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Correspondence to Asgar Jamneshan.

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Jamneshan, A., Kupper, M. & Streckfuß, M. Measures and Integrals in Conditional Set Theory. Set-Valued Var. Anal 26, 947–973 (2018). https://doi.org/10.1007/s11228-018-0478-3

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  • DOI: https://doi.org/10.1007/s11228-018-0478-3

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