A note on liftings of linear continuous functionals
HTML articles powered by AMS MathViewer
- by Horst Osswald PDF
- Proc. Amer. Math. Soc. 120 (1994), 453-456 Request permission
Abstract:
We show that for each bounded Loeb space $(\Lambda ,{L_\nu }(\mathfrak {A}),\hat \nu )$ a functional $\varphi \in {L_\infty }{(\Lambda )’}$ has a lifting if and only if $\varphi \in {L_1}(\Lambda )$. If $p \in [1,\infty [$, then every $\varphi \in {L_p}{(\Lambda )’}$ has a lifting.References
- Sergio Albeverio, Raphael Høegh-Krohn, Jens Erik Fenstad, and Tom Lindstrøm, Nonstandard methods in stochastic analysis and mathematical physics, Pure and Applied Mathematics, vol. 122, Academic Press, Inc., Orlando, FL, 1986. MR 859372 R. A. Anderson, A nonstandard representation of Brownian motion and Ito-integration, Israel J. Math. 15 (1976).
- Edwin Hewitt and Karl Stromberg, Real and abstract analysis, Graduate Texts in Mathematics, No. 25, Springer-Verlag, New York-Heidelberg, 1975. A modern treatment of the theory of functions of a real variable; Third printing. MR 0367121
- Douglas N. Hoover and Edwin Perkins, Nonstandard construction of the stochastic integral and applications to stochastic differential equations. I, II, Trans. Amer. Math. Soc. 275 (1983), no. 1, 1–36, 37–58. MR 678335, DOI 10.1090/S0002-9947-1983-0678335-1
- Albert E. Hurd and Peter A. Loeb, An introduction to nonstandard real analysis, Pure and Applied Mathematics, vol. 118, Academic Press, Inc., Orlando, FL, 1985. MR 806135
- H. Jerome Keisler, An infinitesimal approach to stochastic analysis, Mem. Amer. Math. Soc. 48 (1984), no. 297, x+184. MR 732752, DOI 10.1090/memo/0297 T. L. Lindstrøm, Hyperfinite stochastic integration. I, II, III, and addendum, Math. Scand. 46 (1980).
- Peter A. Loeb, Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), 113–122. MR 390154, DOI 10.1090/S0002-9947-1975-0390154-8
- Peter A. Loeb, A functional approach to nonstandard measure theory, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 251–261. MR 737406, DOI 10.1090/conm/026/737406
- Horst Osswald, Vector valued Loeb measures and the Lewis integral, Math. Scand. 68 (1991), no. 2, 247–268. MR 1129592, DOI 10.7146/math.scand.a-12360
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 120 (1994), 453-456
- MSC: Primary 03H05; Secondary 28E05, 46S20
- DOI: https://doi.org/10.1090/S0002-9939-1994-1165064-6
- MathSciNet review: 1165064