Markov Processes and Stochastic Calculus

• Ω: An abstract space of points ω ∈ Ω. • F : a σ-field (or σ-algebra) on Ω. That is, a collection of subsets of Ω satisfying: 1. Ω ∈ F . 2. Let A ⊆ Ω such that A ∈ F then Ac = Ω−A ∈ F . 3. Let A1, A2, A3, · · · ∈ F then A1 ∪A2 ∪A3 ∪ · · · ∈ F . The elements of F are called events. Condition (1) above simply states that the space Ω is necessarily an event. Conditions (2) and (3) state that the collection of events is closed under the set operations of complement and countable union.