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Probability Theory Cheat Sheet (DRAFT) by

It's about measure theory and Probability theory I use Achim Klenkes Masterclass Probability theory

This is a draft cheat sheet. It is a work in progress and is not finished yet.

Measure Theory

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.

Set Functions

Let A ⊂ 2Ω and let μ : A → [0, ∞] be a set function. We say that μ is
monoton
μ(A) ≤ μ(B) for any two sets A, B ∈ A with A ⊂ B,
additiv
if μ ( n⊎ i=1 Ai ) = n∑ i=1 μ(Ai) for any choice of finitely many mutually disjoint sets A1, . . . , An ∈ A with n⋃ i=1 Ai ∈ A,
σ-additive
if μ ( ∞⊎ i=1 Ai ) = ∞∑ i=1 μ(Ai) for any choice of countably many mu- tually disjoint sets A1, A2, . . . ∈ A with ∞⋃ i=1 Ai ∈ A,
subadd­itive
if for any choice of finitely many sets A, A1, . . . , An ∈ A with A ⊂ n⋃ i=1 Ai, we have μ(A) ≤ n∑ i=1 μ(Ai), and
σ-suba­dditive
if for any choice of countably many sets A, A1, A2, . . . ∈ A with A ⊂ ∞⋃ i=1 Ai, we have μ(A) ≤ ∞∑ i=1 μ(Ai)
Let A be a semiring and let μ : A → [0, ∞] be a set function with μ(∅) = 0. μ is called a
content
if μ is additive,
premeasure
if μ is σ-addi­tive,
measure
if μ is a premeasure and A is a σ-algebra
probab­ility measure
if μ is a measure and μ(Ω) = 1
Let A be a semiring. A content μ on A is called
finite
if μ(A) < ∞ for every A ∈ A
σ-finite
if there exists a sequence of sets Ω1, Ω2, . . . ∈ A such that Ω = ∞⋃ n=1 Ωn and such that μ(Ωn) < ∞ for all n ∈ N.
Examples
Dirac measure
Let ω ∈ Ω and δω (A) = 1A(ω) . Then δω is a probab­ility measure on any σ-algebra A ⊂ 2Ω . δω is called the
uniform distri­bution
Let Ω be a finite nonempty set. By μ(A) := #A #Ω for A ⊂ Ω, we define a probab­ility measure on A = 2Ω