Abstract
We shall prove lifting theorems for Pettis measurable stochastic processes and Pettis integrable functions on Loeb spaces with values in nonseparable locally convex spaces. We apply the results to give a nonstandard proof of
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[1]
Vitali’s convergence theorem for uniformly Pettis integrable functions with values in weakly complete spaces.
We will show for functions with weakly compact range:
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[2]
the existence of conditional expectations in adabted Loeb spaces,
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[3]
Keisler’s Fubini theorem,
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[4]
the existence of solutions to special cases of nondeterministic vector valued Peano-Caratheodory differential equations.
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References
S. Albeverio, J. E. Fenstad, R. Høegh Krohn, T. L. Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics; Academic Press (1986)
R. A. Anderson, A nonstandard representation of Brownian motion and Ito-integration; Israel J. Math. 15 (1976)
N. J. Cutland, Nonstandard measure theory and its application; Bull. London Math. Soc. 15 (1983)
J. Diestel, J. Uhl, Progress in vector measures 1977 — 83; Lecture notes in Math. 1033 (1983)
R. F. Geitz, Pettis integration; Proc. Amer. Math. Soc. 82 (1983)
S. Heinrich, Ultrapowers of locally convex spaces; Mathematische Nachrichten 118 (1984)
C. W. Henson, L. C. Moore, The nonstandard theory of topological vector spaces; Trans. Amer. Math. Soc. 172 (1972)
A. E. Hurd, P. A. Loeb, An introduction to nonstandard real analysis; Academic Press (1985)
H. J. Keisler, An infinitesimal approach to stochastic analysis; Memoirs Amer. Math Soc. (1984)
P. A. Loeb, Conversion from standard to nonstandard measure spaces and applications to probability theory; Trans. Amer. Mat. Soc. 211 (1975)
P. A. Loeb, A functional approach to nonstandard measure theory; Contemporary Math. 26 (1984)
P. A. Loeb, H. Osswald, Nonstandard integration theory on topological vector spaces; to appear
W. A. J. Luxemburg, A general theory of monads; Applications of model theory to algebra, analysis and probability, Holt Rinehart and Winston, New York (1969)
W. R. Pestman, Convergence of martingales with values in locally convex Suslin spaces; Math. Institut, Rijsuniversiteit Groningen, preprint (1981)
B. J. Pettis, On integration on vector spaces; Trans. Amer. Math. Soc. 44 (1936)
D. A. Ross, Lifting theorems in nonstandard measure theory; Proc. Amer. Math. Soc. 109 (1990)
M. Talagrand, Pettis integrability; Memoirs Amer. Math. Soc (1984)
G. Winkler, Choquet order and simplices; Springer Lecture notes in Math. 1145 (1986).
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Osswald, H. (1995). A Nonstandard Approach to the Pettis Integral. In: Albeverio, S.A., Luxemburg, W.A.J., Wolff, M.P.H. (eds) Advances in Analysis, Probability and Mathematical Physics. Mathematics and Its Applications, vol 314. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8451-7_6
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DOI: https://doi.org/10.1007/978-94-015-8451-7_6
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