Classes of sets |
Definition1.1(σ-algebra):A class of sets A ⊂ 2Ω if it fulfils the following three conditions: (i) Ω ∈ A. (ii) A is closed under complements. (iii) A is closed under countable unions. |
Defitinition1.2(algebra):A class of sets A ⊂ 2Ω is called an algebra if the following three conditions are fulfilled: (i) Ω ∈ A. (ii) A is \-closed. (iii) A is ∪-closed |
Definition 1.3(ring):A class of sets A ⊂ 2Ω is called a ring if the following three condi- tions hold: (i) ∅ ∈ A. (ii) A is \-closed. (iii) A is ∪-closed. A ring is called a σ-ring if it is also σ-∪-closed |
Definition 1.4(semiring): A class of sets A ⊂ 2Ω is called a semiring if (i) ∅ ∈ A, (ii) for any two sets A, B ∈ A the difference set B \ A is a finite union of mutually disjoint sets in A, (iii) A is ∩-closed. |
Definition 1.5(Dynkin-system):A class of sets A ⊂ 2Ω is called a λ-system (or Dynkin’s λ-system) if (i) Ω ∈ A, (ii) for any two sets A, B ∈ A with A ⊂ B, the difference set B \ A is in A, and (iii) ⊎∞ n=1 An ∈ A for any choice of countably many pairwise disjoint sets A1, A2, . . . ∈ A. |
Definition 1.6 (liminf & limsup):Let A1, A2, . . . be subsets of Ω. The sets lim inf n→∞ An := ∞⋃ n=1 ∞⋂ m=n Am and lim sup n→∞ An := ∞⋂ n=1 ∞⋃ m=n Am are called limes inferior and limes superior, respectively, of the sequence (An)n∈N. |
Theorem 1.1(Intersection of classes of sets):Let I be an arbitrary index set, and assume that Ai is a σ-algebra for every i ∈ I. Hence the intersection AI := {A ⊂ Ω : A ∈ Ai for every i ∈ I} = ⋂ i∈I Ai is a σ-algebra. The analogous statement holds for rings, σ-rings, algebras and λ- systems. However, it fails for semirings |
Theorem 1.2 (Generated σ-algebra):Let E ⊂ 2Ω . Then there exists a smallest σ-algebra σ(E) with E ⊂ σ(E): σ(E) := ⋂ A⊂2Ω is a σ-algebra A⊃E A. |
Theorem 1.3(∩-closed λ-system):Let D ⊂ 2Ω be a λ-system. Then D is a π-system ⇐⇒ D is a σ-algebra. |
Theorem 1.4(Dynkin’s π-λ theorem): If E ⊂ 2Ω is a π-system, then σ(E) = δ(E). |
Definition 1.7(Topology): Let Ω = ∅ be an arbitrary set. A class of sets τ ⊂ Ω is called a topology on Ω if it has the following three properties: (i) ∅, Ω ∈ τ . (ii) A ∩ B ∈ τ for any A, B ∈ τ . (iii) (⋃ A∈F A) ∈ τ for any F ⊂ τ . The pair (Ω, τ ) is called a topological space. The sets A ∈ τ are called open, and the sets A ⊂ Ω with Ac ∈ τ are called closed. |
Definition 1.8(Borel σ-algebra):Let (Ω, τ ) be a topological space. The σ- algebra B(Ω) := B(Ω, τ ) := σ(τ ) that is generated by the open sets is called the Borel σ-algebra on Ω. The elements A ∈ B(Ω, τ ) are called Borel sets or Borel measurable sets. |
Definition 1.8 (Trace of a class of sets):Let A ⊂ 2Ω be an arbitrary class of subsets of Ω and let A ∈ 2Ω \ {∅}. The class A∣ ∣A := {A ∩ B : B ∈ A} ⊂ 2A is called the trace of A on A or the restriction of A to A. |
