On mixing in infinite measure spaces

Summary Two concepts of mixing for null-preserving transformations are introduced, both coinciding with (strong) mixing if there is a finite invariant measure. The authors believe to offer the correct answer to the old problem of defining mixing in infinite measure spaces. A sequence of sets is called semiremotely trivial if every subsequence contains a further subsequence with trivial remote -algebra (=tail -field). A transformation T is called mixing if (T ^–n A) is semiremotely trivial for every set A of finite measure; completely mixing if this is true for every measurable A. Thus defined mixing is exactly the condition needed to generalize certain theorems holding in finite measure case. For invertible non-singular transformations complete mixing implies the existence of a finite equivalent invariant mixing measure. If no such measure exists, complete mixing implies that for any two probability measures ₁,₂, in total variation norm.Research of this author is supported by the National Science Foundation (U.S.A.) under grant GP 7693.     

On non-Archimedean hilbertian spaces

We consider a special class of non-Archimedean Banach spaces, called Hilbertian, for which every one-dimensional linear subspace has an orthogonal complement. We prove that all immediate extensions of c_o, contained in l^∞, are Hilbertian. In this way we construct examples of Hilbertian spaces over a non-spherically complete valued field without an orthogonal base.     

On grand Lebesgue spaces on sets of infinite measure

We consider grand Lebesgue spaces on sets of infinite measure and study the dependence of these spaces on the choice of the so‐called. We also consider Mikhlin and Marcinkiewicz theorems on Fourier multipliers in the setting of grand spaces.     

On the non-Archimedean metric Mahler measure

Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted M_∞, and prove that M_∞(α)=1 if and only if α is a root of unity. We further show that M_∞ defines a projective height on as a vector space over Q. Finally, we demonstrate how to compute M_∞(α) when α is a surd.     

On convergence of information in spaces with infinite invariant measure

Summary The paper extends the ergodic theorems of information theory (Shannon-MacMillan-Breiman theorems) to spaces with an infinite invariant measure. An L ₁ difference theorem and a pointwisa ratio theorem are proved, for the information of spreading partitions. For the validity of the theorems it is assumed that the supremum f ^* of the conditional information given the increasing past is integrable. Simple necessary and sufficient conditions for the integrability of f ^* are obtained in special cases: If the initial partition is composed of one state of a null-recurrent Markov chain, then f ^* is integrable if and only if the partition of this state according to the first return times has finite entropy.Research of both authors was supported by the National Science Foundation (U. S. A.), under Grant GP 7693.     

On multiparameter ergodic and martingale theorems in infinite measure spaces

Summary A unified proof is given of several ergodic and martingale theorems in infinite measure spaces.The research of this author is in part supported by the National Science Foundation, grant MCS-8301619     

On local properties of non-Archimedean analytic spaces II

The non-Archimedean analytic spaces are studied. We extend to the general case notions and results defined earlier only for strictly analytic spaces. In particular, we prove that any strictly analytic space admits a unique rigid model.     

On relations between non-Archimedean power series spaces

The power series spaces of finite type, A₁(α), and infinite type, A_∞(α), are the most known and important examples of non-Archimedean nuclear Fréchet spaces. We study when Aπ (α) has a subspace (or quotient) isomorphic to A_q(b).     

On the ergodic theory of non-integrable functions and infinite measure spaces

We prove a Chow-Robbins type result for an ergodic, non-negative SSSP, and a similar result for transformations preserving infinite measure, which implies that for these transformations, no “absolute” version of Hopf's theorem can hold.     

On the entropy of a product endomorphism in infinite measure spaces

Summary In this note, a relationship is established between the entropy, defined by Krengel for an endomorphism of a -finite measure space, and the notion of a spreading partition. This relationship is used to answer in the quasi-finite case a question raised by Krengel concerning the entropy of the product endomorphism on the direct product of a finite and -finite measure space.     

Selectors in non-Archimedean spaces

Continuous selectors on the hyperspace F(X) are studied, when X is a non-Archimedean space. It is shown that a non-Archimedean space has a continuous selector if and only if it is topologically well orderable. Another characterization is given in terms of density and complete metrizability.     

On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

The paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field.     

On the Mazur-Ulam theorem in non-Archimedean fuzzy normed spaces

We prove that the Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy normed space.     

Borel hierarchies in infinite products of Polish spaces

Let H be a product of countably infinite number of copies of an uncountable Polish space X. Let Σ_ξ be the class of Borel sets of additive class ξ for the product of copies of the discrete topology on X (the Polish topology on X), and let . We prove in the Lévy-Solovay model that

Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Bornological spaces of non-Archimedean valued functions

Let C(X,E) be the space of all continuous functions from an ultraregular space X to a non-Archimedean locally convex space E. Necessary and/or sufficient conditions are given so that C(X,E), with the topology of uniform convergence on compact sets or with the topology of simple convergence, is bornological or c-ultrabornological.