The Converse of the Individual Ergodic Theorem
نویسنده
چکیده
converge almost everywhere to a finite limit f*(x). It then follows that the limit function/* is integrable and that/*(7x) =/*(x) almost everywhere. This result can be applied to certain cases in which the given measure m is not preserved by the transformation T. In order to discuss this application, we recall some terminology for measures and transformations. If (X, S) is a measurable space, and if p and m are two measures on S, we shall say that p is absolutely continuous with respect to m, and shall write p<£.m, if m(E) =0 implies p(E) =0. The measures p and m are said to be equivalent if both p<^m and m<üp, in which case we write p = m. If T is a measurable transformation on a measure space (X, S, m), then T defines a second measure on S, denoted by mT~l and defined by mT~1(E)—m(T~lE). To say that T is measure-preserving is to say that m = mT~1. A transformation T will be called an absolutely continuous transformation if mT~1<^m, and will be called nonsingular if mT~1 = m. Nonsingular transformations are sometimes called measurability preserving transformations. A measurable transformation T is called incompressible if m(E—T~1E)=0 implies m(T~1E — E)=Q. In suggestive language, T is incompressible if and only if ECT~lE a.e. implies E=T~XE a.e. It is clear that T is incompressible if and only if m(T~1E — E)=0 implies m(E— T~XE) =0, and that an incompressible transformation is always nonsingular. A measure-preserving transformation on a finite measure space is incompressible, but this need not be the case
منابع مشابه
Individual ergodic theorem for intuitionistic fuzzy observables using intuitionistic fuzzy state
The classical ergodic theory hasbeen built on σ-algebras. Later the Individual ergodictheorem was studied on more general structures like MV-algebrasand quantum structures. The aim of this paper is to formulate theIndividual ergodic theorem for intuitionistic fuzzy observablesusing m-almost everywhere convergence, where m...
متن کاملA Mean Ergodic Theorem For Asymptotically Quasi-Nonexpansive Affine Mappings in Banach Spaces Satisfying Opial's Condition
متن کامل
Non-linear ergodic theorems in complete non-positive curvature metric spaces
Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...
متن کامل. L O ] 1 2 Ju n 20 12 RANDOMNESS AND NON - ERGODIC SYSTEMS
We characterize the points that satisfy Birkhoff’s ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Löf random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the...
متن کاملThe Uniform Mean-Square Ergodic Theorem for Wide Sense Stationary Processes
It is shown that the uniform mean-square ergodic theorem holds for the family of wide sense stationary sequences, as soon as the random process with orthogonal increments, which corresponds to the orthogonal stochastic measure generated by means of the spectral representation theorem, is of bounded variation and uniformly continuous at zero in a mean-square sense. The converse statement is also...
متن کاملON LOCAL HUDETZ g-ENTROPY
In this paper, a local approach to the concept of Hudetz $g$-entropy is presented. The introduced concept is stated in terms of Hudetz $g$-entropy. This representation is based on the concept of $g$-ergodic decomposition which is a result of the Choquet's representation Theorem for compact convex metrizable subsets of locally convex spaces.
متن کامل