On infinite products of non-Archimedean measure spaces

We study generalized ‘probabilistic measures’ taking values in non-Archimedean fields (in particular, fields of p-adic numbers). We prove the theorem on the existence of probability on a product of non-Archimedean probabilistic spaces.     

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 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 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 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.     

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 data depth in infinite dimensional spaces

The concept of data depth leads to a center-outward ordering of multivariate data, and it has been effectively used for developing various data analytic tools. While different notions of depth were originally developed for finite dimensional data, there have been some recent attempts to develop depth functions for data in infinite dimensional spaces. In this paper, we consider some notions of depth in infinite dimensional spaces and study their properties under various stochastic models. Our analysis shows that some of the depth functions available in the literature have degenerate behaviour for some commonly used probability distributions in infinite dimensional spaces of sequences and functions. As a consequence, they are not very useful for the analysis of data satisfying such infinite dimensional probability models. However, some modified versions of those depth functions as well as an infinite dimensional extension of the spatial depth do not suffer from such degeneracy and can be conveniently used for analyzing infinite dimensional data.     

On p-adic vector measure spaces

For a separating algebra R of subsets of a set X, E a complete Hausdorff non-Archimedean locally convex space and m:R→E a bounded finitely additive measure, we study some of the properties of the integrals with respect to m of scalar-valued functions on X. The concepts of convergence in measure, with respect to m, and of m-measurable functions are introduced and several results concerning these notions are given.     

Measurable rigidity of actions on infinite measure homogeneous spaces, II

We consider the problems of measurable isomorphisms and joinings, measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces. Our main focus is on systems of algebraic origin: actions of lattices and other discrete subgroups on homogeneous spaces where is a sufficiently rich unimodular subgroup in a semi-simple group . We also consider actions of discrete groups of isometries of a pinched negative curvature space , acting on the space of horospheres . For such systems we prove that the only measurable isomorphisms, joinings, quotients, etc., are the obvious algebraic (or geometric) ones. This work was inspired by the previous work of Shalom and Steger but uses completely different techniques which lead to more general results.

     

Operator renewal theory and mixing rates for dynamical systems with infinite measure

We develop a theory of operator renewal sequences in the context of infinite ergodic theory. For large classes of dynamical systems preserving an infinite measure, we determine the asymptotic behaviour of iterates L ⁿ of the transfer operator. This was previously an intractable problem.Examples of systems covered by our results include (i) parabolic rational maps of the complex plane and (ii) (not necessarily Markovian) nonuniformly expanding interval maps with indifferent fixed points.In addition, we give a particularly simple proof of pointwise dual ergodicity (asymptotic behaviour of \(\sum_{j=1}^{n}L^{j}\)) for the class of systems under consideration.In certain situations, including Pomeau-Manneville intermittency maps, we obtain higher order expansions for L ⁿ and rates of mixing. Also, we obtain error estimates in the associated Dynkin-Lamperti arcsine laws.     

On the congruence problem in infinite dimensional sesquilinear spaces

If two subspaces V and V of a sesquilinear space E are congruent (i.e., there is an isometry : E E with (V)=V) then their corresponding quadratic lattices V(V, E) and V(V, E) are isomorphic. It is shown that the converse holds for important types of sesquilinear spaces E, provided that dim(E) ₃. However, the converse generally fails if dim(E) ₃.     

On compact subsets in coechelon spaces of infinite order

For coechelon spaces of infinite order it is proved that every compact subset of is contained in a closed absolutely convex hull of some null sequence if and only if the matrix is regularly decreasing.

     

On inductive limits of measure spaces

We introduce a category of (topological) measure spaces in which inductive limitis exist and where the Banach spaces and (1≤p≤+∞) are isometric for arbitrary inductive systems of (topological) measure spaces.     

On the geometry of metric measure spaces

We introduce and analyze lower (Ricci) curvature bounds  ⩾ K for metric measure spaces . Our definition is based on convexity properties of the relative entropy regarded as a function on the L ₂-Wasserstein space of probability measures on the metric space . Among others, we show that  ⩾ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds,  ⩾ K if and only if  ⩾ K for all . The crucial point is that our lower curvature bounds are stable under an appropriate notion of D-convergence of metric measure spaces. We define a complete and separable length metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ⩽ C is closed under D-convergence. Moreover, the family of normalized metric measure spaces with doubling constant ⩽ C and diameter ⩽ L is compact under D-convergence.