测度论

Introduction

0.ConventionsandNotation

1.Notation:Euclideanspace

2.Operationsinvolving

3.Inequalitiesandinclusions

4.Aspaceanditssubsets

5.Notation:generationofclassesofsets

6.Productsets

7.Dotnotationforanindexset

8.Notation:setsdefinedbyconditionsonfunctions

9.Notation:openandclosedsets

10.Limitofafunctionatapoint

11.Metricspaces

12.Standardmetricspacetheorems

13.Pseudometricspaces

I.OperationsonSets

I.Unionsandintersections

2.Thesymmetricdifferenceoperator

3.Limitoperationsonsetsequences

4.Probabilisticinterpretationofsetsandoperationsonthem

II.ClassesofSubsetsofaSpace

1.Setalgebras

2.Examples

3.Thegenerationofsetalgebras

4.TheBorelsetsofametricspace

5.Productsofsetalgebras

6.Monotoneclassesofsets

III.SetFunctions

1.Setfunctiondefinitions

2.Extensionofafinitelyadditivesetfunction

3.Productsofsetfunctions

4.Heuristicsonaalgebrasandintegration

5.Measuresandintegralsonacountablespace

6.Independenceandconditionalprobability(preliminarydiscussion)

7.Dependenceexamples

8.Inferiorandsuperiorlimitsofsequencesofmeasurablesets

9.Mathematicalcounterpartsofcointossing

10.Setwiseconvergenceofmeasuresequences

11.Outermeasure

12.OutermeasuresofcountablesubsetsofR

13.Distanceonasetalgebradefinedbyasubadditivesetfunction

14.Thepseudometricspacedefinedbyanoutermeasure

15.Nonadditivesetfunctions

IV.MeasureSpaces

1.Completionofameasurespace(S,$,λ)

2.GeneralizationoflengthonR

3.Ageneralextensionproblem

4.Extensionofameasuredefinedonasetalgebra

5.ApplicationtoBorelmeasures

6.StrengtheningofTheorem5whenthemetricspaceSiscompleteandseparable

7.Continuitypropertiesofmonotonefunctions

8.ThecorrespondencebetweenmonotoneincreasingfunctionsonRandmeasuresonB(R)

9.DiscreteandcontinuousdistributionsonR

10.Lebesgue-StieltjesmeasuresonR/vandtheircorrespondingmonotonefunctions

11.Productmeasures

12.ExamplesofmeasuresonRN

13.Marginalmeasures

14.Cointossing

15.TheCaratheodorymeasurabilitycriterion

16.Measurehulls

V.MeasurableFunctions

1.Functionmeasurability

2.Functionmeasurabilityproperties

3.Measurabilityandsequentialconvergence

4.Bairefunctions

5.Jointdistributions

6.Measuresonfunction(coordinate)space

……

VI.Intezration

VII.HilbertSoace

VIII.ConvergenceofMeasureSequences

IX.SignedMeasures

X.MeasuresandFunctionsofBoundedVariationonR

XI.ConditionalExpectations;MartingaleTheory

Notation

Index

　　本书是Springer研究生数学丛书之一，对测度论的讲述完全不同于一般的教程。该书将概率论作为测度论的必不缺少的一部分，所以书中的许多例子都是来自概率论，如独立性、马尔科夫过程、条件期望这些都作为本书的组成部分而不是将其置于附录中作为补充。特别是对Sigma代数做了较多的研究，而不是拿来即用。运用伪度量而不是度量给出了集合空间和函数空间的距离更直观的定义。