Measure-theoretic quantifiers and Haar measure

The Haar Measure

In this section, we give a brief review of the measure theory which will be used in later sections. We use [R, Chapters 1 and 2] as our main resource. A σ-algebra on a set X is a collectionM of subsets of X such that ∅ ∈M, if S ∈M, then X \ S ∈ M, and if a countable collection S1, S2, . . . ∈ M, then ∪i=1Si ∈ M. That is, M is closed under complements and countable unions, and contains the empty...

Markov-Kakutani Theorem and Haar Measure

Measure Theoretic Probability

Preface In these notes we explain the measure theoretic foundations of modern probability. The notes are used during a course that had as one of its principal aims a swift introduction to measure theory as far as it is needed in modern probability, e.g. to define concepts as conditional expectation and to prove limit theorems for martingales. Everyone with a basic notion of mathematics and prob...

Measure-Theoretic Probability I

Measure-theoretic Uniformity

Here we present the principal ideas and results of [5] with some indications of proof. We introduce the notion of measure-theoretic uniformity, and we describe its use in recursion theory, hyperarithmetic analysis, and set theory. In recursion theory we show that the set of all sets T such that the ordinals recursive in T are the recursive ordinals has measure 1. In set theory we obtain all of ...