Martingales and Harmonic Analysis

A function f : Ω→ R is called F -measurable if f−1(B) := {f ∈ B} := {ω ∈ Ω : f(ω) ∈ B} ∈ F for all Borel sets B ⊆ R. Denote by F 0 the collection of sets in F with finite measure, i.e., F 0 := {E ∈ F : μ(E) <∞}. The measure space (Ω,F , μ) is called σ-finite if there exist sets Ei ∈ F 0 such that ⋃∞ i=0Ei = Ω. If needed, these sets may be chosen to additionally satisfy either (a) Ei ⊆ Ei+1 or (b) Ei∩Ej = ∅ whenever i 6= j. Part (a) follows by taking E′ i := ⋃i j=0Ei, and part (b) by setting E ′′ i := E ′ i \E′ i−1, where E′ −1 := ∅. Unless otherwise stated, it is always assumed in the sequel that (Ω,F , μ) is σ-finite. An F -measurable function f : Ω→ R is called σ-integrable if it is integrable on all sets of finite measure, i.e., if 1Ef ∈ L(F , μ) for all E ∈ F . Denote the collection of all such functions by Lσ(F , μ).