Introduction In 12] is presented a constructive deenition of Borel subsets of Cantor space and their inclusion relation. They are seen as symbolic expressions built from the Boolean algebra of simple (closed and open) subsets of by formal countable disjunction and conjunction. If X is such a symbolic expression, it is clear for instance how to deene by induction the formal complement X 0 of X: ...

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued $f$ on ${\mathbb R^n}$. This article devoted investigating the mapping properties of new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well its VMO CMO subspace...

In this paper, we fully characterize the duality mapping over space of matrices that are equipped with Schatten norms. Our approach is based on analysis saturation Hölder inequality for We prove in our main result that, p∈(1,∞), real-valued Schatten-p norm a continuous and single-valued function provide an explicit form its computation. For special case p = 1, set-valued; by adding rank constra...

Uniqueness of Gibbs states and decay properties of averaged, two point correlation functions are proved for many-body potentials in continuum statistical mechanical models via Dobrushin uniqueness techniques. Introduction Gross [6], using Dobrushin uniqueness techniques [1], has given decay rates for two point correlation functions in classical lattice models at high temperature or low activity...

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will no...

We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simultaneously.

It is shown that a measurable function from an atomless Loeb probability space (Ω,A, P ) to a Polish space is at least continuum-to-one valued almost everywhere. It follows that there is no injective mapping h : [0, 1] → Ω such that h([a, b]) is Loeb measurable for each 0 ≤ a < b ≤ 1 and P (h([0, 1])) > 0. Thus, when an atomless Loeb measurable algebra on an internal set of cardinality continuu...