Mathematical Foundations of Probability Theory

TL;DR

This paper explores the mathematical foundations of probability theory through measure theory, emphasizing a rigorous, functional analysis-based approach that prepares readers for advanced topics like stochastic processes and statistical theory.

Contribution

It provides a comprehensive, pure mathematical perspective on probability theory rooted in measure theory, linking it to functional analysis and future stochastic analysis.

Findings

Establishes probability theory within measure theory and functional analysis.

Prepares the groundwork for stochastic processes and statistical theory.

Offers a continuously updated, rigorous mathematical framework.

Abstract

In the footsteps of the book \textit{Measure Theory and Integration By and For the Learner} of our series in Probability Theory and Statistics, we intended to devote a special volume of the very probabilistic aspects of the first cited theory. The book might have assigned the title : From Measure Theory and Integration to Probability Theory. The fundamental aspects of Probability Theory, as described by the keywords and phrases below, are presented, not from experiences as in the book \textit{A Course on Elementary Probability Theory}, but from a pure mathematical view based on Measure Theory. Such an approach places Probability Theory in its natural frame of Functional Analysis and constitutes a firm preparation to the study of Random Analysis and Stochastic processes. At the same time, it offers a solid basis towards Mathematical Statistics Theory. The book will be continuously updated and improved on a yearly basis.