Nearly Borel sets and product measures

Direct Product Testing With Nearly Identical Sets

In this work we analyze a direct product test in which each of two provers receives a subset of size n of a ground set U , |U | = Θ(n), and the two subsets intersect in about (1 − δ)n elements. We show that if each of the provers provides labels to the n elements it received, and the labels of the two provers agree in the intersection between the subsets with non-negligible probability, then th...

Measures of Sets Partitioning Borel ' S Simply Normal Numbers

5 = {a) : 0 < a) < 1, lim pm (00) = 1/2}, m -> co then v(S) = 1. Borel s arguments in support of this impressive fact were flawed, but valid proofs were supplied by later workers (see [1]). The set S defines those numbers in [0, 1] which are said to be simply normal to base 2. The very definition of simply normal numbers induces rather natural families of partitions of [0, 1]. Motivated by the ...

Loeb Measures and Borel Algebras

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...

Borel Sets and Countable Models

We show that certain families of sets and functions related to a countable structure A are analytic subsets of a Polish space. Examples include sets of automorphisms, endomorphisms and congruences of A and sets of the combinatorial nature such as coloring of countable plain graphs and domino tiling of the plane. This implies, without any additional set-theoretical assumptions, i.e., in ZFC alon...

A Course on Borel Sets