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\author{LuciaZhang}
\pdfinfo{
  /Title (appendix-b-measure-and-probability-theory.pdf)
  /Creator (Cheatography)
  /Author (LuciaZhang)
  /Subject (Appendix B+ - Measure \& Probability Theory Cheat Sheet)
}

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\noindent
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\begin{tabulary}{11cm}{L}
    \vspace{-2pt}\large{\bf{\textcolor{DarkBackground}{\textrm{Appendix B+ - Measure \& Probability Theory Cheat Sheet}}}} \\
    \normalsize{by \textcolor{DarkBackground}{LuciaZhang} via \textcolor{DarkBackground}{\uline{cheatography.com/136048/cs/48168/}}}
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  \mymulticolumn{2}{p{5.377cm}}{\bf\textcolor{white}{Cheatographer}}  \\
  \vspace{-2pt}LuciaZhang \\
  \uline{cheatography.com/luciazhang} \\
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  \mymulticolumn{1}{p{5.377cm}}{\bf\textcolor{white}{Cheat Sheet}}  \\
   \vspace{-2pt}Not Yet Published.\\
   Updated 6th June, 2026.\\
   Page {\thepage} of \pageref{LastPage}.
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  %\includegraphics[width=48px,height=48px]{dave.jpeg}
  Measure your website readability!\\
  www.readability-score.com
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\begin{tabularx}{5.377cm}{x{1.9908 cm} x{2.9862 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{σ-algebras \& Borel sets}}  \tn
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A ⊆ 2\textasciicircum{}Ω\textasciicircum{} is a σ-algebra & if Ω ∈ A, A ∈ A ⇒ A\textasciicircum{}c\textasciicircum{} ∈ A, and A\textasciitilde{}i\textasciitilde{} ∈ A ⇒ ⋃\textasciitilde{}i∈N\textasciitilde{} A\textasciitilde{}i\textasciitilde{} ∈ A \tn 
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σ(E) & intersection of all σ-algebras containing E \tn 
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ℬ(V) & σ(open subsets of V) \tn 
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\begin{tabularx}{5.377cm}{x{1.59264 cm} x{3.38436 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Measures}}  \tn
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Measure\{\{width=35\}\} & μ(∅)=0 and μ(⋃ A\textasciitilde{}i\textasciitilde{})=Σ μ(A\textasciitilde{}i\textasciitilde{}) for disjoint A\textasciitilde{}i\textasciitilde{} \tn 
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Probability & P(Ω)=1 \tn 
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Product & (⊗\textasciitilde{}j\textasciitilde{}μ\textasciitilde{}j\textasciitilde{})(A\textasciitilde{}1\textasciitilde{} × ... × A\textasciitilde{}n\textasciitilde{})=∏\textasciitilde{}j\textasciitilde{} μ\textasciitilde{}j\textasciitilde{}(A\textasciitilde{}j\textasciitilde{}) \tn 
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Lebesgue & λ\textasciicircum{}d\textasciicircum{}(∏\textasciitilde{}j\textasciitilde{} (a\textasciitilde{}j\textasciitilde{},b\textasciitilde{}j\textasciitilde{})) = ∏\textasciitilde{}j\textasciitilde{} (b\textasciitilde{}j\textasciitilde{}-a\textasciitilde{}j\textasciitilde{}) \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Measurability}}  \tn
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Measurable\{\{width=33\}\} & f\textasciicircum{}-1\textasciicircum{}(A\textasciitilde{}2\textasciitilde{}) ∈ 𝒜\textasciitilde{}1\textasciitilde{} for all A\textasciitilde{}2\textasciitilde{} ∈ 𝒜\textasciitilde{}2\textasciitilde{} \tn 
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Check on generators & 𝒜\textasciitilde{}2\textasciitilde{}=σ(ℰ), enough to check\{\{nl\}\} f\textasciicircum{}-1\textasciicircum{}(E) ∈ A\textasciitilde{}1\textasciitilde{} for E ∈ ℰ \tn 
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Strongly measurable & f\textasciitilde{}n\textasciitilde{} simple, f\textasciitilde{}n\textasciitilde{} → f pointwise or μ-a.e. \tn 
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Pettis & f strongly measurable ⇔ f separably valued and ⟨f,v′⟩ measurable for every v′ ∈ V \tn 
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\begin{tabularx}{5.377cm}{x{1.69218 cm} x{3.28482 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Bochner integration, change of variables}}  \tn
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μ-simple\{\{width=33\}\} & f = Σ\textasciitilde{}j=1\textasciitilde{}\textasciicircum{}n\textasciicircum{} 1\textasciitilde{}A\_j\textasciitilde{} v\textasciitilde{}j\textasciitilde{}, μ(A\textasciitilde{}j\textasciitilde{}) \textless{} ∞ \tn 
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 & ∫ f dμ = Σ μ(A\textasciitilde{}j\textasciitilde{}) v\textasciitilde{}j\textasciitilde{} for simple f \tn 
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Bochner definition & f\_n→f μ-a.e. and ∫ ||f-f\_n|| dμ→0 \tn 
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Bochner criterion & f strongly μ-measurable, ∫ ||f|| dμ \textless{} ∞ \tn 
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Norm bound & ||∫ f dμ|| ≤ ∫ ||f|| dμ \tn 
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Duality & ⟨∫ f dμ, v′⟩ = ∫ ⟨f,v′⟩ dμ \tn 
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L\textasciicircum{}p\textasciicircum{} & μ-a.e. equivalence classes of strongly μ-measurable functions with finite L\textasciicircum{}p\textasciicircum{} norm \tn 
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dν/dμ & density of ν w.r.t μ, ν ≪ μ \tn 
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Pushforward & T\textasciitilde{}\#\textasciitilde{} μ(A) = μ(T\textasciicircum{}-1\textasciicircum{}(A))\{\{nl\}\} ∫ f d(T\textasciitilde{}\#\textasciitilde{}μ) = ∫ f∘T dμ \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Banach-valued RVs}}  \tn
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Random variable\{\{width=30\}\} & X:(Ω,A)→(V,B(V)) measurable \tn 
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\mymulticolumn{2}{x{5.377cm}}{P\textasciitilde{}X\textasciitilde{} = X\textasciitilde{}\#\textasciitilde{}P, so P{[}X ∈ B{]}=P\textasciitilde{}X\textasciitilde{}(B)} \tn 
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\mymulticolumn{2}{x{5.377cm}}{σ(X) = \{X\textasciicircum{}-1\textasciicircum{}(B): B ∈ ℬ(V)\}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{E{[}X{]} = ∫\textasciitilde{}Ω\textasciitilde{} X(ω)dP(ω), if ∫ ||X|| dP \textless{} ∞} \tn 
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\mymulticolumn{2}{x{5.377cm}}{E{[}φ(X){]} = ∫\textasciitilde{}Ω\textasciitilde{} φ(X(ω))dP(ω) = ∫\textasciitilde{}V\textasciitilde{} φ(v)dP\textasciitilde{}X\textasciitilde{}(v)} \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Conditional probability \& independence}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{P{[}A|B{]} = P{[}A∩B{]}/P{[}B{]}, P{[}B{]}\textgreater{}0} \tn 
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\mymulticolumn{2}{x{5.377cm}}{A,B independent ⇔ P{[}A∩B{]}=P{[}A{]}P{[}B{]}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{X\textasciitilde{}i\textasciitilde{} independent ⇔ σ(X\textasciitilde{}i\textasciitilde{}) independent} \tn 
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\mymulticolumn{2}{x{5.377cm}}{If X\textasciitilde{}1\textasciitilde{},...,X\textasciitilde{}n\textasciitilde{} independent and integrable: E{[}∏X\textasciitilde{}i\textasciitilde{}{]}=∏E{[}X\textasciitilde{}i\textasciitilde{}{]}} \tn 
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\begin{tabularx}{5.377cm}{x{2.4885 cm} x{2.4885 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Conditional expectation}}  \tn
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For simple Y = Σ 1\textasciitilde{}A\_j\textasciitilde{} y\textasciitilde{}j\textasciitilde{} \{\{width=35\}\} & E`{[}X|Y{]}`(ω) = 1/P{[}A\textasciitilde{}j\textasciitilde{}{]}∫\textasciitilde{}A\_j\textasciitilde{}X dP for ω ∈ A\textasciitilde{}j\textasciitilde{} \tn 
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Z=E{[}X|ℱ{]} & iff Z is ℱ-measurable and \{\{nl\}\}∫\textasciitilde{}B\textasciitilde{}Z dP=∫\textasciitilde{}B\textasciitilde{} X dP for all B ∈ F \tn 
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\mymulticolumn{2}{x{5.377cm}}{E{[}X|Y{]} := E{[}X|σ(Y){]}, P{[}A|Y{]} := E{[}1\textasciitilde{}A\textasciitilde{}|σ(Y){]}} \tn 
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\begin{tabularx}{5.377cm}{x{1.4931 cm} x{3.4839 cm} }
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Regular conditional distributions \& Doob-Dynkin}}  \tn
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\{\{width=35\}\}Goal: & define P{[}X ∈ B | Y=y{]} even when P{[}Y=y{]}=0 \tn 
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\mymulticolumn{2}{x{5.377cm}}{κ\textasciitilde{}X|F\textasciitilde{}: Ω × ℬ(V) → {[}0,1{]}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{For A∈ℱ, B∈ℬ(V): P{[}A∩\{X∈B\}{]} = ∫\textasciitilde{}A\textasciitilde{} κ\textasciitilde{}X|F\textasciitilde{}(ω,B)dP(ω)} \tn 
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Doob-Dynkin & κ σ(Y)-measurable ⇔ κ = τ∘Y for measurable τ \tn 
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\mymulticolumn{2}{x{5.377cm}}{τ\textasciitilde{}X|Y\textasciitilde{}(y,B) := P{[}X∈B | Y=y{]}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Conditional densities}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{f\textasciitilde{}Y\textasciitilde{}(y) = ∫\textasciitilde{}R\textasciicircum{}m\textasciitilde{} f\textasciitilde{}X,Y\textasciitilde{}(x,y) dx} \tn 
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\mymulticolumn{2}{x{5.377cm}}{f\textasciitilde{}X|Y\textasciitilde{}(x|y) = f\textasciitilde{}X,Y\textasciitilde{}(x,y)/f\textasciitilde{}Y\textasciitilde{}(y), for f\textasciitilde{}Y\textasciitilde{}(y)\textgreater{}0} \tn 
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\mymulticolumn{2}{x{5.377cm}}{P{[}X ∈ B | Y=y{]} = ∫\textasciitilde{}B\textasciitilde{} f\textasciitilde{}X|Y\textasciitilde{}(x|y) dx} \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Gaussians}}  \tn
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X∼N(μ,σ$^{\textrm{2}}$): \{\{width=20\}\} & f\textasciitilde{}X\textasciitilde{}(x) = 1/√(2πσ$^{\textrm{2}}$) exp(−(x−μ)$^{\textrm{2}}$/ (2σ$^{\textrm{2}}$)) \tn 
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X∼N(μ,Σ): & f\textasciitilde{}X\textasciitilde{}(x) ∝ det(Σ)\textasciicircum{}-1/2\textasciicircum{} exp(−1/2 (x−μ)\textasciicircum{}T\textasciicircum{}Σ\textasciicircum{}-1\textasciicircum{}(x−μ)) \tn 
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Gaussian facts & marginals Gaussian; conditionals Gaussian; jointly Gaussian + uncorrelated ⇔ independent \tn 
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\mymulticolumn{2}{x{5.377cm}}{\bf\textcolor{white}{Distances \& Divergences}}  \tn
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\mymulticolumn{2}{x{5.377cm}}{D\textasciitilde{}TV\textasciitilde{}(P,Q) = sup\textasciitilde{}A∈𝒜\textasciitilde{}|P(A)−Q(A)|} \tn 
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\mymulticolumn{2}{x{5.377cm}}{D\textasciitilde{}H\textasciitilde{}(P,Q) = (1/√2)(∫(√(dP/dμ)−√(dQ/dμ))$^{\textrm{2}}$ dμ)\textasciicircum{}1/2\textasciicircum{}} \tn 
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\mymulticolumn{2}{x{5.377cm}}{D\textasciitilde{}KL\textasciitilde{}(P||Q) = ∫ log(dP/dQ)dP if P≪Q; ∞ otherwise} \tn 
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Expectation bound & ||E\textasciitilde{}P\textasciitilde{}{[}f{]} − E\textasciitilde{}Q\textasciitilde{}{[}f{]}|| ≤ 2D\textasciitilde{}H\textasciitilde{}(P,Q) (E\textasciitilde{}P\textasciitilde{}||f||$^{\textrm{2}}$+E\textasciitilde{}Q\textasciitilde{}||f||$^{\textrm{2}}$)\textasciicircum{}1/2\textasciicircum{} \tn 
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% That's all folks
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