Related papers: A Proof of the Ergodic Theorem using Nonstandard A…

Our goal in the present paper is to give a new ergodic proof of a well-known Veech's result, build upon our previous works.

We give a simple proof of the Fourier Inversion Theorem, using the methods of nonstandard analysis.

We give a proof of a Martingale Representation Theorem using the methods of nonstandard analysis.

A very short and direct proof along the lines of the Kamae-Katznelson-Weiss approach.

We extend the theory of ergodic optimization and maximizing measures to the non-commutative field of C*-dynamical systems. We then provide a result linking the ergodic optimizations of elements of a C*-dynamical system to the convergence of…

Using nonstandard analysis, an intuitive and very short proof of the Radon-Nikodym theorem is provided

We prove the ergodic Closing Lemma for Nonsingular Endomorphisms.

We present a short proof of Szemer\'edi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity.

We give a proof of the uniform convergence of Fourier series, using the methods of nonstandard analysis.

We prove a version of pointwise Ergodic Theorem for non-stationary random dynamical systems. Also, we discuss two specific examples where the result is applicable: non-stationary iterated function systems and non-stationary random matrix…

Recently, T. Tao gave a finitary proof a convergence theorem for multiple averages with several commuting transformations and soon later, T. Austin gave an ergodic proof of the same result. Although we give here one more proof of the same…

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.

In this paper The Ergodic Hypothesis is proven for one class of functions defined in the infinite dimensional unite cube where is given an action of some semigroup of mappings without the condition on metric transitivity. The result has not…

We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

In the first part of the paper the natural scheme for proving noncommutative individual ergodic theorems for multiple sequences is described and applied to obtain results on unrestricted convergence of multiaverages. In the second part…

We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…

Using nonstandard analysis, a very short and elementary proof of the Doob-Meyer decomposition and the Dol{\'e}ans Dade theorems is provided.

This is a survey on Sarnak's Conjecture

We apply the methods of ergodic theory to both simplify and significantly extend some classical results due to Stewart, Tijdeman, and Ruzsa. One of the notable features of our approach is the utilization of pointwise ergodic theory.