Measure and integration

TL;DR

This paper provides an introduction to measure theory, integration, and function spaces, emphasizing explicit integral computations and applications in probability, including discrete, continuous, and quantum contexts.

Contribution

It offers a comprehensive, standard presentation of measure theory and integration with detailed motivations and applications, including an advanced quantum probability discussion.

Findings

Explicit computation techniques for integrals

Connections between measure theory and probability

Applications to quantum probability

Abstract

This is an introduction to measure theory, integration and function spaces, with all the needed preliminaries included, and with some applications included as well. We first discuss some basic motivations, coming from discrete probability, that we develop in detail, as a preliminary to general measure theory. Then we discuss measure theory, integration and function spaces, all developed in a standard way, and with emphasis on the explicit computation of various integrals. Finally, we come back to probability, discrete and continuous, with a more advanced discussion, of quantum flavor.