Sigma Algebras

"A probability model begins by deciding which questions are allowed to have probabilities."

Overview

Sigma algebras define the measurable events, observations, and information structure on which probability and learning objectives are built.

Measure theory is the grammar behind rigorous probability. Earlier probability chapters taught how to compute with random variables and distributions. This chapter explains what those objects are when sample spaces are infinite, events are generated by observations, and densities depend on a base measure.

This section uses LaTeX Markdown throughout. Inline mathematics uses $...$, and display mathematics uses `

`. The focus is the foundation needed for ML: expected loss, pushforward distributions, convergence of estimators, likelihood ratios, importance sampling, KL divergence, and support mismatch.

Prerequisites

• Sets and Logic
• Functions and Mappings
• Introduction to Probability and Random Variables
• Adversarial Game Theory

Companion Notebooks

Learning Objectives

After completing this section, you will be able to:

• Define algebras, sigma algebras, measurable spaces, and measurable maps
• Construct generated sigma algebras from finite generators
• Explain why countable closure is required for limits and probability
• Identify Borel sigma algebras on real vector spaces
• Use pullbacks to model observations, features, and random variables
• Build product sigma algebras for vector-valued and sequence-valued data
• Separate events from arbitrary subsets of the sample space
• Explain why measurability is a prerequisite for probability claims
• Connect information partitions to model observability
• Prepare for Lebesgue integration by identifying measurable functions

Study Flow

1. Read the pages in order and pause after each page to restate the main definition or theorem.
2. Run theory.ipynb when you want to check the formulas numerically.
3. Use exercises.ipynb after the reading path, not before it.
4. Return to this overview page when you need the chapter-level navigation.

Runnable Companions

• Theory notebook
• Exercises notebook