Introduction to Stochastic Processes

We use the notation A ∈ F to indicate that the set A is in F . Remark that since A∩B = (A ∪B) and more generally ⋂ Ai = ( ⋃ Ai) c (Exercise: Prove this equality) axiom ii) holds with union replaced by intersection. Also we have A−B = A ∩B so if A,B ∈ F then A−B ∈ F . Basically you should think of a σ-algebra as a system of subsets in which you can perform any of the usual set-theoretic operations (union, intersection, difference) on countably many sets. We get an obvious example by taking F to be the system of all subsets of Ω, P(Ω). At the other extreme the system consisting only of Ω itself and ∅ is a σ-algebra. Given any system of subsets S of Ω we can consider the smallest σ-algebra containing S. This is the intersection of all the σ-algebras containing S. Since the set of all subsets of Ω is a σ-algebra containing S, there is at least one σ-algebra containing S. The smallest σ-algebra containing S is called the σ-algebra generated by S.