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\author{Julia22}
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  /Title (probability-theory.pdf)
  /Creator (Cheatography)
  /Author (Julia22)
  /Subject (Probability Theory Cheat Sheet)
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    {\parbox{\dimexpr\textwidth-2\fboxsep\relax}{\noindent
        \hspace*{-6pt}\includegraphics[width=5.8cm]{/web/www.cheatography.com/public/images/cheatography_logo.pdf}}
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    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Probability Theory Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{Julia22} via \textcolor{DarkBackground}{\uline{cheatography.com/191398/cs/39786/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}Julia22 \\
  \uline{cheatography.com/julia22} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 15th August, 2023.\\
   Page {\thepage} of \pageref{LastPage}.
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  Measure your website readability!\\
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\begin{multicols*}{3}

\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Measure Theory}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\bf{Classes of sets}}} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition1.1}}({\bf{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.} \tn 
% Row Count 6 (+ 5)
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Defitinition1.2}}({\bf{algebra}}):A class of sets A ⊂ 2Ω is called an algebra if the following three conditions are fulfilled: (i) Ω ∈ A. (ii) A is \textbackslash{}-closed. (iii) A is ∪-closed} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.3}}({\bf{ring}}):A class of sets A ⊂ 2Ω is called a ring if the following three condi- tions hold: (i) ∅ ∈ A. (ii) A is \textbackslash{}-closed. (iii) A is ∪-closed. A ring is called a {\bf{σ-ring}} if it is also σ-∪-closed} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.4}}({\bf{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 \textbackslash{} A is a finite union of mutually disjoint sets in A, (iii) A is ∩-closed.} \tn 
% Row Count 20 (+ 5)
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.5}}({\bf{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 \textbackslash{} A is in A, and (iii) ⊎∞ n=1 An ∈ A for any choice of countably many pairwise disjoint sets A1, A2, . . . ∈ A.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.6}} ({\bf{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.} \tn 
% Row Count 33 (+ 6)
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\begin{tabularx}{5.377cm}{X}
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Measure Theory (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Theorem 1.1}}({\bf{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} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Theorem 1.2}} ({\bf{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.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Theorem 1.3}}({\bf{∩-closed λ-system}}):Let D ⊂ 2Ω be a λ-system. Then D is a π-system ⇐⇒ D is a σ-algebra.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Theorem 1.4}}({\bf{Dynkin's π-λ theorem}}): If E ⊂ 2Ω is a π-system, then σ(E) = δ(E).} \tn 
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% Row 11
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.7}}({\bf{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 {\bf{topological space}}. The sets A ∈ τ are called open, and the sets A ⊂ Ω with Ac ∈ τ are called closed.} \tn 
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.8}}({\bf{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.} \tn 
% Row Count 31 (+ 6)
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\mymulticolumn{1}{x{5.377cm}}{\bf\textcolor{white}{Measure Theory (cont)}}  \tn
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\mymulticolumn{1}{x{5.377cm}}{{\emph{Definition 1.8}} ({\bf{Trace of a class of sets}}):Let A ⊂ 2Ω be an arbitrary class of subsets of Ω and let A ∈ 2Ω \textbackslash{} \{∅\}. The class A∣ ∣A := \{A ∩ B : B ∈ A\} ⊂ 2A is called the trace of A on A or the restriction of A to A.} \tn 
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\begin{tabularx}{5.377cm}{x{1.64241 cm} x{3.33459 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Set Functions}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{Let A ⊂ 2Ω and let μ : A → {[}0, ∞{]} be a set function. We say that μ is} \tn 
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monoton & μ(A) ≤ μ(B) for any two sets A, B ∈ A with A ⊂ B, \tn 
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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, \tn 
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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, \tn 
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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 \tn 
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% Row 5
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\seqsplit{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) \tn 
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% Row 6
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\mymulticolumn{2}{x{5.377cm}}{Let A be a semiring and let μ : A → {[}0, ∞{]} be a set function with μ(∅) = 0. μ is called a} \tn 
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content & if μ is additive, \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Set Functions (cont)}}  \tn
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premeasure & if μ is σ-additive, \tn 
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measure & if μ is a premeasure and A is a σ-algebra \tn 
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probability measure & if μ is a measure and μ(Ω) = 1 \tn 
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\mymulticolumn{2}{x{5.377cm}}{Let A be a semiring. A content μ on A is called} \tn 
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finite & if μ(A) \textless{} ∞ for every A ∈ A \tn 
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σ-finite & if there exists a sequence of sets Ω1, Ω2, . . . ∈ A such that Ω = ∞⋃ n=1 Ωn and such that μ(Ωn) \textless{} ∞ for all n ∈ N. \tn 
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\mymulticolumn{2}{x{5.377cm}}{Examples} \tn 
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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 \tn 
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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 \tn 
% Row Count 25 (+ 5)
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% That's all folks
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