1 Sets, Fields, Measures, and Probability Measures Definition Let Ω be a probability space. We call ω ∈ Ω a sample point and S ⊂ Ω an event. Definition A class F of subsets of Ω is called a field or algebra if: • Ω ∈ F • If A ∈ F then A ∈ F . • If A,B ∈ F then A ∪B ∈ F . (This is called finite additivity.) The class is a σ-field or a σ-algebra if the following condition holds as well: • If A1, A2, ... ∈ F then ⋃∞ i=1 Ai ∈ F as well. (This is called countable additivity.) An set that is an element of F is called an F-set, and is said to be measurable F . The σ-field generated by a class of sets, A, σ(A), is the intersection of all σ-fields that contain A. Definition The extended real line is [−∞,∞]; it includes both positive and negative infinity. Definition Let R be the σ-field generated by the bounded rectangles [x = (x1, ..., xk) : ai ≤ xi ≤ bi, i = 1, ..., k] ⊂ R. The elements of R are called the k-dimensional Borel sets. Note that R contains all the open and closed sets (among other things). R is sometimes written as B. Theorem 1.1 If A is a class of sets in Ω and Ω0 ⊂ Ω, let A ∩ Ω0 = [A ∩ Ω0 : A ∈ A]. If F is a σ-field in Ω then F ∩ Ω0 is a σ-field in Ω0. If A generates the σ-field F in Ω then A ∩ Ω0 generates the σ-field F ∩ Ω0 in Ω0. That is, σ(A ∩ Ω0) = σ(A) ∩ Ω0. Definition A class P of subsets of Ω is a π-system if whenever A,B ∈ P, A ∩B ∈ P as well.