The Kolmogorov continuity theorem, Hölder continuity, and the Kolmogorov-Chentsov theorem

Gaussian Processes: the Wiener Isometry and the Kolmogorov-chentsov Theorem

We will be most interested in Gaussian processes indexed by an interval J ⇢ R, usually J = [0,1) or J = [0, T ] for some 0 < T < 1. Under weak hypotheses there will always exist centered Gaussian processes with specified covariance functions, but it is not generally the case that such processes will have versions with continuous sample paths. Our primary objective in this chapter will be to dev...

A new proof for the Banach-Zarecki theorem: A light on integrability and continuity

To demonstrate more visibly the close relation between thecontinuity and integrability, a new proof for the Banach-Zareckitheorem is presented on the basis of the Radon-Nikodym theoremwhich emphasizes on measure-type properties of the Lebesgueintegral. The Banach-Zarecki theorem says that a real-valuedfunction $F$ is absolutely continuous on a finite closed intervalif and only if it is continuo...

Uniqueness Theorem for Ordinary Differential Equations with Hölder Continuity

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...

Krieger’s Finite Generator Theorem for Actions of Countable Groups Iii

We continue the study of Rokhlin entropy, an isomorphism invariant for p.m.p. actions of countable groups introduced in Part I. In this paper we prove a non-ergodic finite generator theorem and use it to establish subadditivity and semi-continuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify...