Source: J. Symbolic Logic
Volume 71, Issue 2
Let T be a positive L₁-L∞ contraction. We
prove that the following statements are equivalent in constructive
The projection in L₂ on the space of invariant functions
- 2. The sequence (Tⁿ)n ∈ N
Cesáro-converges in the L₂ norm;
- 3. The sequence (Tⁿ)n ∈ N
Cesáro-converges almost everywhere.
Thus, we find necessary and sufficient conditions for the Mean Ergodic
Theorem and the Dunford-Schwartz Pointwise Ergodic Theorem.
As a corollary we obtain a constructive ergodic theorem for ergodic
measure-preserving transformations. This answers a question posed by Bishop.
Full-text: Access denied (no subscription
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Jeremy Avigad and Ksenija Simic, Fundamental notions of analysis in subsystems of second-order arithmetic, Annals of Pure and Applied Logic, to appear.
Errett Bishop, Mathematics as a numerical language, Intuitionism and Proof Theory (Proceedings of the summer conference at Buffalo, N.Y., 1968), North-Holland, Amsterdam,1970, pp. 53--71.
Mathematical Reviews (MathSciNet): MR270894
Errett Bishop and Douglas Bridges, Constructive analysis, Grundlehren der Mathematischen Wissenschaften, vol. 279, Springer-Verlag,1985.
Mathematical Reviews (MathSciNet): MR804042
Errett A. Bishop, Foundations of constructive analysis, McGraw-Hill Publishing Company, Ltd.,1967.
Mathematical Reviews (MathSciNet): MR221878
N. Dunford and J. T. Schwartz, Linear operators. Part I: General theory, Interscience Publishers,1958.
Hajime Ishihara and Luminiţa Vîţǎ, Locating subsets of a normed space, Proceedings of the American Mathematical Society, vol. 131 (2003), no. 10, pp. 3231--3239.
Ulrich Krengel, Ergodic theorems, Studies in Mathematics, de Gruyter,1985.
Mathematical Reviews (MathSciNet): MR797411
J. A. Nuber, A constructive ergodic theorem, Transactions of the American Mathematical Society, vol. 164 (1972), pp. 115--137.
Mathematical Reviews (MathSciNet): MR291411
--------, Erratum to `A constructive ergodic theorem', Transactions of the American Mathematical Society, vol. 216 (1976), p. 393.
Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press,1983.
Mathematical Reviews (MathSciNet): MR833286
Bas Spitters, Constructive and intuitionistic integration theory and functional analysis, Ph.D. thesis, University of Nijmegen,2002.
Peter Walters, An introduction to ergodic theory, Graduate Texts in Mathematics, vol. 79, Springer-Verlag,1982.
Mathematical Reviews (MathSciNet): MR648108