Probability Theory

3. Zero-One Laws

3.1 Almost Sure Events

Definition 3.1 (Almost Sure Event): Let A∈F\evA \in \sigmaF. We say that A\evA occurs almost surely or a.s. if P ⁣(A)=1\probP{\evA} = 1.
Note: This notion can be exteneded to any set A⊂ΩA \subset \Omega which is not necessarily an event, i.e. A∉FA \not\in \sigmaF might be possible. We say that AA occurs almost surely if there exists an event A′∈F\evA' \in \sigmaF such that A′⊂A\evA' \subset A and P ⁣(A′)=1\probP{\evA'} = 1.

Since random variables are often defined up to a null set, the notion of almost sure occurence is particularly usefule when we manipulate random variables. For example if X,YX, Y are two random variables, we write X≤Y  a.s.X \leq Y \sep \as if P ⁣(X≤Y)=1\probP{X \leq Y} = 1.

Example (Single point in continuous rv): Let U∼U⁡([0,1])U \sim \lawU([0,1]). For every x∈[0,1]x \in [0,1] we have U≠x  a.s.U \neq x \sep \as since P ⁣(U≠x)=P ⁣(U∈[0,1]∖{x})=λ([0,1]∖{x})=1 \probP{U \neq x} = \probP{U \in [0,1] \setminus \set{x}} = \lambda([0,1] \setminus \set{x}) = 1
Proposition 3.2 (Countable Intersection of a.s. events): Let (An)n∈I(\evA_n)_{n \in \setI} be a collection of events, where I\setI is a finite or countable index set. If for all n∈In \in \setI the event An\evA_n occurs almost surely, then the intersection ⋂n∈IAn\bigcap_{n \in \setI} \evA_n also occurs almost surely.