Lower and upper Loeb-integrals
HTML articles powered by AMS MathViewer
- by D. Landers and L. Rogge PDF
- Trans. Amer. Math. Soc. 358 (2006), 3263-3283 Request permission
Abstract:
We introduce the concepts of lower and upper Loeb-integrals for an internal integration structure. These are concepts which are similarly useful for Loebs internal integration theory as the concepts of inner and outer Loeb-measures for Loebs measure theory.References
- J. M. Aldaz, A modified functional approach to nonstandard measure theory, Caribbean J. Math. Comput. Sci. 3 (1993), no. 1-2, 17–20. MR 1366716
- Nigel Cutland (ed.), Nonstandard analysis and its applications, London Mathematical Society Student Texts, vol. 10, Cambridge University Press, Cambridge, 1988. Papers from a conference held at the University of Hull, Hull, 1986. MR 971063, DOI 10.1017/CBO9781139172110
- Klaus Floret, Maß- und Integrationstheorie, Teubner Studienbücher Mathematik. [Teubner Mathematical Textbooks], B. G. Teubner, Stuttgart, 1981 (German). Eine Einführung. [An introduction]. MR 609518, DOI 10.1007/978-3-322-93106-1
- 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
- D. Landers and L. Rogge, Universal Loeb-measurability of sets and of the standard part map with applications, Trans. Amer. Math. Soc. 304 (1987), no. 1, 229–243. MR 906814, DOI 10.1090/S0002-9947-1987-0906814-1
- Dieter Landers and Lothar Rogge, Nonstandard methods for families of $\tau$-smooth probability measures, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1151–1156. MR 954998, DOI 10.1090/S0002-9939-1988-0954998-8
- Dieter Landers and Lothar Rogge, Nichtstandard Analysis, Springer-Lehrbuch. [Springer Textbook], Springer-Verlag, Berlin, 1994 (German, with German summary). MR 1275833, DOI 10.1007/978-3-642-57915-8
- 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, Weak limits of measures and the standard part map, Proc. Amer. Math. Soc. 77 (1979), no. 1, 128–135. MR 539645, DOI 10.1090/S0002-9939-1979-0539645-6
- 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
- Peter A. Loeb, A nonstandard functional approach to Fubini’s theorem, Proc. Amer. Math. Soc. 93 (1985), no. 2, 343–346. MR 770550, DOI 10.1090/S0002-9939-1985-0770550-9
- Washek F. Pfeffer, Integrals and measures, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 42, Marcel Dekker, Inc., New York-Basel, 1977. MR 0460580
- Sommers, U., Theorie unendlicher Loeb-Maße, Diplomarbeit, Duisburg, 1998.
Additional Information
- D. Landers
- Affiliation: Mathematisches Institut der Universität zu Köln, Weyertal 86, D–50931 Köln, Germany
- Email: landers@mi.uni-koeln.de
- L. Rogge
- Affiliation: Fachbereich Mathematik der Gerhard-Mercator-Universität GHS Duisburg, Lotharstrasse 65, D–47048 Duisburg, Germany
- Email: rogge@math.uni-duisburg.de
- Received by editor(s): October 5, 2001
- Published electronically: March 24, 2006
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 358 (2006), 3263-3283
- MSC (2000): Primary 28E05; Secondary 26E35
- DOI: https://doi.org/10.1090/S0002-9947-06-04042-6
- MathSciNet review: 2218975