The Strong Law of Large Numbers

15.1 This section gives some fundamental definitions in the theory of probability, such as the definitions of a probability space and a random variable.15.2 In this section the fundamental concept of independence is developed.15.3 We give an example of singular function, which arises naturally from probability theory.15.4 Given Ω, set Ω _n = Ω for all n ∈ N and ΩN = Π _n Ω _n . The one-sided shift transformation v on ΩN is defined by (x _n ) _{n ≥1} ↦ (x _{n +i}) _{n ≥i}. The probability of a v-invariant event is either 0 or 1 (Proposition 15.4.2) and v is μ-ergodic (Proposition 15.4.3)—see Section 11.4. Then, using Birkhoff’s ergodic theorem, we prove the strong law of large numbers (Theorem 15.4.1).15.5 A number x ≠ m/bn is said to be completely normal if, for every integer k and every k-tuple (u₁,…, u _k ) of base-b digits, the k-tuple appears in the base-b expansion of x with asymptotic relative frequency 1/bk. Almost every number (for Lebesgue measure) in [0, 1] is completely normal (Proposition 15.5.2).