Probability Spaces

The first step in the FE-540 notes is to give random experiments a mathematical home. A probability space is a triple \((\Omega,\mathcal F,\mathbb P)\): \(\Omega\) is the sample space of outcomes, \(\mathcal F\) is the collection of events we are allowed to measure, and \(\mathbb P\) assigns their probabilities.

The restriction to a sigma-algebra matters when the sample space is infinite. It contains \(\Omega\), is closed under complements, and is closed under countable unions. On the real line, the Borel sigma-algebra is generated by open intervals. A measurable space \((\Omega,\mathcal F)\) is therefore the structure that lets a probability measure talk about events without pretending every subset is measurable.

The notes use unions, intersections, complements, and De Morgan's laws repeatedly. They are the set operations that make later statements about random variables and stochastic processes precise.