Let (Ω, F, P) be a probability space. To recapitulate: fi is a set called the “sample space”. Its elements are called sample points.F is a σ-field of subsets of Ω containing fi itself. Elements of F are called events. P is a probability measure (i.e., a countably additive nonnegative measure with total mass 1) on the measurable space (Ω, F). If an event A is of the type A = ω ∈ Ω R (ω) for some property R (.), we may write P(R) for P(A). An event is called a sure event if P(A) = 1 and a null event if P(A) = 0. Alternatively, R (.) is said to hold almost surely (a.s. for short) if P(R) = 1. Many statements in probability theory are made with the qualification “almost surely”, though this may not always be stated explicitly.
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