Measure-theoretic Probability Theory

Graduate-level Probability Theory

Probability Measures

Definitions

Field

A class \(\displaystyle \mathcal{F}\) of subsets of \(\displaystyle \Sigma\) is called a field if it contains \(\displaystyle \Sigma\) and is closed under complements and finite unions.

• \(\displaystyle \Sigma \in F\)
• \(\displaystyle A \in F \implies A^c \in F\)
• \(\displaystyle A, B \in F \implies A \cup B \in F\)

Sigma-field

A class F is called a \(\displaystyle \sigma\)-field if it is also closed under countable unions

• \(\displaystyle A_1, A_2, ... \in F \implies A_1 \cup A_2 \cup ... \in F\)

Probability measure space

Given a sigma field F on space \(\displaystyle \Sigma\) and probability measure P,
\(\displaystyle (\Sigma, F, P)\) is a probability measure space.

Textbooks

• [Probability and Measure by Patrick Billingsley]