Math - Stat - 491 - Fall 2014 - Notes

2.1 σ-field and Probability measure A sample space Ω and a collection F of subsets from Ω subject to the following conditions: 1. Ω ∈ F . 2. If A ∈ F , then its complement A ∈ F . 3. If A1, A2, ... is a finite or countably infinite sequence of subsets from F , then ⋃ Ai ∈ F . Any collection F satisfying these postulates is termed a σ-field or σ-algebra. Two immediate consequences of the definitions are that the empty set ∅ ∈ F and that if A1, A2, ... is a finite or countably infinite sequence of subsets from F , then ⋂ iAi = ( ⋃ iA c i ) c ∈ F . The sample space Ω may be thought of as the set of possible outcomes of an experiment. Points ω in Ω are called sample outcomes, realizations, or elements. Subsets of Ω are called Events. Example. If we toss a coin twice then Ω = {HH,HT, TH, TT}. The event that the first toss is heads is A = {HH,HT}. A probability measure or distribution μ on the events in F should satisfy the properties: