Topic 
Textbook Section 
1. Events and Classes of Events 


outcomes, events, review of set theory 
1.2 

limits of events 
1.3, 1.4 

πclasses, fields and σfields 
1.5, 1.6 

Borel σfields 
1.7, 1.8 

Dynkin's πλ class theorem 
2.2 
2. Probability Measures and Measures 


probability measures and general measures 
2.1 – 2.3 

properties of measures, probability distributions, uniqueness 
2.1 

distribution functions, LebesgueStieltjes measures 


extension and existence theorems 
2.4, 2.5 
3. Random Variables and Measurability 


random variables and measurable functions, σfield generated by a random variable 
3.1 – 3.3 

induced measures, distribution of a random variable 
3.2 

sufficient conditions for measurability 
3.2, 5.1 
4. Expectation and Integration 


definitions, consistency of definitions 
5.1, 5.2, 5.4 

properties of expectation and integrals 
5.2 

LebesgueStieltjes integration, absolute continuity, Riemann integrals 
5.6 

monotone and dominated convergence theorems, extensions 
5.2, 5.3 
5. Advanced Integration 


equating differently defined integrals 
5.5, 5.6 

moments and inequalities, finiteness of moments 
5.2 

product spaces, multiple integration, TonelliFubini theorem 
5.7 – 5.9 

characteristic functions, inversion formulas 
9.1 – 9.5 
6. Independence 


independence of finitely many events or random variables 
4.1, 4.2 

infinite collections of independent events or random variables 
4.3, 4.4 

BorelCantelli lemmas, tail events, Kolmogorov's 01 law 
4.5 
7. Convergence of Sequences of Random Variables 


modes of convergence and their relationships 
6.1 – 6.3, 6.5 

moment inequalities, Jensen's inequality 
6.5 

uniform integrability, convergence in mean 
6.5, 6.6 
8. The Law of Large Numbers and Convergence of Random Series 


strong law of large numbers 
7.1, 7.4 

GlivenkoCantelli theorem 
7.5 

convergence of series 
7.3, 7.6 
9. Convergence of Distributions 


weak convergence, Scheffé's theorem 
8.1, 8.2, 8.5 

Slutsky's theorem, delta method, Skorohod's lemma, continuous mapping 
8.3, 8.6 

tightness, Prohorov's theorem, the continuity theorem 
9.5, 9.6 

portmanteau theorem, multivariate convergence 
8.4 
10. Weak Convergence for Sums and Maxima 


central limit theorem, LindebergFeller and Lyapunov theorems 
9.7, 9.8 

convergence to types, infinitely divisible and stable distributions 
8.7 

extreme value distributions 
8.7 
11. Absolute Continuity and Conditional Expectation 


conditional expectation 
10.2, 10.3 

regular conditional distributions 


conditional independence and exchangeablilty 


absolute continuity, RadonNikodym theorem, Lebesgue decomposition 
10.1 