Measure-theoretic chaos

Prequential probability: game-theoretic = measure theoretic

This article continues study of the prequential framework for evaluating a probability forecaster. Testing the hypothesis that the sequence of forecasts issued by the forecaster is in agreement with the observed outcomes can be done using prequential notions of probability. It turns out that there are two natural notions of probability in the prequential framework: game-theoretic, whose idea go...

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 ...

Measure-theoretic unfriendly colorings

We consider the problem of finding a measurable unfriendly partition of the vertex set of a locally finite Borel graph on standard probability space. After isolating a sufficient condition for the existence of such a partition, we show how it settles the dynamical analog of the problem (up to weak equivalence) for graphs induced by free, measurepreserving actions of groups with designated finit...