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Contents
Preface- please read!
xi A Question of Terminology
Xlll A Guide to Notation
xiv Chapter O: A Branching-Process Exalnple
1
0.0. Introductory remarks. 0.1. Typical number of children, X. 0.2. Size
of n th generation, Z,,. 0.3. Use of conditional expectations. 0.4. Extinction
probability, *r. 0.5. Pause for thought: measure. 0.6. Our first martingale.
e ' 0.7. Convergence (or not) of xpectatlons. 0.8. Finding the distribution of
M. 0.9. Concrete example.
PART A: FOUNDATIONS
Chapter 1: Measure Spaces
1.0. Introductory remarks. 1.1. Definitions of algebra, a-algebra. 1.2. Ex-
amples. Borel a-algebras, B($), B = B(R). 1.3. Definitions concerning
set functions. 1.4. Definition of measure space. 1.5. Definitions con-
cerning measures. 1.6. Lemma. Uniqueness of extension, ,r-systems. 1.7.
Theorem. Carathdodory's extension theorem. 1.8. Lebesgue measure Leb
on ((0, 1], B(0, 1]). 1.9. Lemma. Elementary
Monotone-convergence properties of measures.
inequalities. 1.10. Lemma. 1.11. Example/Warning.
Chapter 2: Events
2.1. Model for experiment:
Examples of (Q,O r') pairs.
lira sup, lim inf, lim, etc.
(12, ', P). 2.2. The
2.4. Almost surely
2.6. Definitions.
23
intuitive meaning. 2.3.
(a.s.) 2.5. Reminder:
limsupE,, (E,,i.o.). 2.7.
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