Foundations of the theory of probability

CONTENTS

Page

EditorsNote. iii

Preface — v

I.

ElementaryTheoryofProbability

§

1. Axioms 2

§

2. Therelation toexperimental data 3

§

3. Notesonterminology 5

§

4. Immediatecorollariesoftheaxioms;conditionalproba-

bilities

;

TheoremofBayes

6

§

5. Independence
   8

§

6. Conditional probabilitiesas randomvariables;Markov

chains 12

II. InfiniteProbabilityFields

§

1. AxiomofContinuity 14

§

2. Borelfieldsofprobability 16

§3. Examplesof infinitefieldsof

probability

18

III. RandomVariables

§

1. Probabilityfunctions 21

§

2. Definitionofrandomvariablesandofdistributionfunc-

tions 22

§3. Multi-dimensionaldistributionfunctions 24

§

4. Probabilitiesininfinite-dimensionalspaces 27

§

5. Equivalentrandomvariables;variouskindsofconverg-

ence 33

IV. MathematicalExpectations

§1. AbstractLebesgueintegrals 37

§

2. Absoluteandconditionalmathematicalexpectations.... 39

§

3. TheTchebycheffinequality 42

§

4. 
   

   Somecriteriaforconvergence

   
   

   43

   
   

§

5. Differentiation
   andintegrationofmathematicalexpecta-

tionswith

respecttoaparameter 44

Vll