Measures and all that --- A Tutorial

TL;DR

This tutorial provides a comprehensive overview of measure theory fundamentals, including Borel sets, measure spaces, integration, product measures, dual spaces, and key theorems, with applications to stochastic models and bisimulations.

Contribution

It offers an accessible, structured introduction to measure theory concepts and their applications in stochastic and logical models, bridging theory and practice.

Findings

Detailed explanation of Borel sets and generators

Introduction to product measures and their applications

Discussion of Radon-Nikodym and Riesz theorems in measure spaces

Abstract

This tutorial gives an overview of some of the basic techniques of measure theory. It includes a study of Borel sets and their generators for Polish and for analytic spaces, the weak topology on the space of all finite positive measures including its metrics, as well as measurable selections. Integration is covered, and product measures are introduced, both for finite and for arbitrary factors, with an application to projective systems. Finally, the duals of the Lp-spaces are discussed, together with the Radon-Nikodym Theorem and the Riesz Representation Theorem. Case studies include applications to stochastic Kripke models, to bisimulations, and to quotients for transition kernels.