1. Basic Definitions - Probability Space (_O_,F,P) ~ Prob prop'ties: 1) P(A)>=0; 2) o'-additive; 3) P(_O_)=1. ~ Inc/Exc rule; sub-additivity; continuity - Fatou's Lemma: P(Ai e.a.) <= lim-inf P(Ai) <= lim-sup P(Ai) <= P(Ai i.o.) - df: F(x)=F(x+); x F(x) <= F(y); F(-oo)=0, F(+oo)=1 2. Dynkin's Theorem * Lambda System* * Pi System* l1: _O_ \in L l2: A^c \in L if A \in L p1: A B \in L if A,B\in L l3: Disjoint ctble unions Thm (ED): a) P a \pi-system in L a \lambda-system => o'(P) \subset L b) P a \pi-system => o'(P) = L(P) 3. Two Constructions i) Discrete: Countable \Omega, \sum{p_i} = 1, => B=2^\Omega okay ii) Continuous: Prob density iii) General 1-d: CDF 4. Constructions of Probability Spaces - Infinite Bernoulli sequence - Cantor distribution? - Big Thm 2.4.3: Ctbl-additive set function on a field F has ! extension to o'(F) 5. Measure Constructions 2 - Lebesgue measure on (0,1] (\lambda_2(dx)) 6. Counter-examples? E.g. Uniform on integers; Finitely-additive measures; ------------ Thu 1/26: Introduce: 1) { w: Xn(w) -> X(w) } = \cap{km} { w: |Xn(w) - X(w)| > 1/k } = \cap{k 1/k } Focus on: 1) Discrete probability spaces 2) Lebesgue measure 3) Continuous probability spaces 4) Cantor Example? Show: 1) Uniform distribution on rationals in [0,1] (what goes wrong?)