The mathematical foundations of probability

A measure-theoretic introduction

Abstractions are there to hide irrelevant things and focus only on the important details. Although it may seem scary sometimes, it is the best tool to manage complexity.

If you ask n mathematicians to define what mathematics is about, you’ll probably get 2n different answers. My definition would be that it is the science of abstracting things away until only the core is left, providing the ultimate framework for reasoning about anything.

Have you ever thought about what probability really is? You have definitely used it to reason about data, do statistical analysis, or even build algorithms to do the reasoning for you by statistical learning. In this post, we are going deep into the rabbit hole and explore the theory of probability with a magnifying glass.

Prerequisites

To follow through, you don’t need any advanced mathematics, I am focusing to explain everything from the grounds up. However, to it is beneficial if you know the following:

• Sets and set operations such as union, intersection and difference.
• Limits and some basic calculus.

Sets and measures