Overview

A Probability Space is the mathematical foundation upon which all of probability theory is built. It provides a consistent framework to quantify “randomness” without falling into logical traps.

In this chapter, we will construct the triplet $(\Omega, \cF, \Pr)$ step-by-step:

1. $\sigma$-Fields: Defining “measureable” events $\cF$.
2. Probability Measures: Defining the function $\Pr$ that assigns probabilities to these events.
3. Carathéodory’s Extension Theorem: A powerful tool to construct probability measures on complex spaces.
4. Lebesgue Measure: The canonical example of a probability measure on continuous space, derived using the extension theorem.