Appendix B+ - Measure & Probability Theory Cheat Sheet (DRAFT) by LuciaZhang

𝒜 ⊆ 2^Ω is a σ-algebra

if Ω ∈ 𝒜, A ∈ 𝒜 ⇒ A^c ∈ 𝒜, and A_i ∈ 𝒜 ⇒ ⋃_i∈N A_i ∈ 𝒜

ℬ(V)

σ(open subsets of V)

Measure

μ(∅)=0 and μ(⋃ A_i)=Σ μ(A_i) for disjoint A_i

Product

(⊗_jμ_j)(A₁ × ... × A_n)=∏_j μ_j(A_j)

Measurable

f^-1(A₂) ∈ 𝒜₁ for all A₂ ∈ 𝒜₂

Strongly measurable

f_n simple, f_n → f pointwise or μ-a.e.

μ-simple

f = Σ_j=1ⁿ 1_{A_j} v_j, μ(A_j) < ∞

Bochner integrable

f_n→f μ-a.e. and ∫ ||f-f_n|| dμ→0

Norm bound

||∫ f dμ|| ≤ ∫ ||f|| dμ

L^p

μ-a.e. equiva­lence classes of strongly μ-meas­urable functions with finite L^p norm

Pushfo­rward

T_# μ(A) = μ(T^-1(A))
∫ f d(T_#μ) = ∫ f∘T dμ

Random variable

X:(Ω,𝒜­)→(­V,B(V)) measurable

σ(X) = {X^-1(B): B ∈ ℬ(V)}

E[φ(X)] = ∫_Ω φ(X(ω)­)dP(ω) = ∫_V φ(v)dP_X(v)

1st Def.

P[A|B] = P[A∩B]­/P[B], P[B]>0

X_i indepe­ndent ⇔ σ(X_i) indepe­ndent

For simple Y = Σ 1_{A_j} y_j

E

[X|Y]

(ω) = 1/P[A_j]∫_{A_j}X dP for ω ∈ A_j

2nd Def.

E[X|Y] := E[X|σ(Y)]
P[A|Y] := E[1_A|σ(Y)]

Goal:

define P[X ∈ B | Y=y] even when P[Y=y]=0

reg. cond. distr. of X given ℱ

P[A∩{X∈B}] = ∫_A κ_X|F(ω,B)d­P(ω), ∀ A∈ℱ, B∈ℬ(V)

reg. cond. distr. of X given Y

P[X∈B | Y=y] := τ_X|Y(y,B) := κ_X|σ(Y)(Y^-1(y),B)

f_Y(y) = ∫_R^m f_X,Y(x,y) dx

P[X ∈ B | Y=y] = ∫_B f_X|Y(x|y) dx

X∼N(μ,σ²):

f_X(x) = 1/√(2πσ²) exp(−(­x−μ)²/ (2σ²))

Gaussian facts

marginals Gaussian; condit­ionals Gaussian; jointly Gaussian + uncorr­elated ⇔ indepe­ndent

D_TV(P,Q) = sup_A∈𝒜|P(A)−­Q(A)|

D_KL(P||Q) = ∫ log(dP­/dQ)dP if P≪Q; ∞ otherwise