测度与范畴学

1．MeasureandCategoryontheLine．，

Countablesets,setsoffirstcategory，nullsets,thetheoremsofCantor．Baire,andBOreI

2．LiouvilleNumbers．

Algebraicandtranscendentalnumbers,measureandcategoryofthesetof

Liouviilehumbers

3．LcbesgueMeasureinr-Space．’

Definitionsandprincipalproperties，measurablesets，theLebesguedensitytheorem

4．ThePropertyofBaire

Itsanalogytomeasurability，propertiesofregularopensets

5．Non-MeasurableSets

Vitalisets,Bernsteinsets，Ulam’stheorem,inaccessiblecardinals,thecon．

tinuumhypothesis

6．TheBanach-MazurGame

Winningstrategies,categoffandlocalcategory,indeterminategames

7．FunctionsofFirstClass．

Oscillation,thelimitofasequenceofcontinuousfunctions,Riemannintegrability

8．TheTheoremsofLusinandEgoroff

ContinuityofmeasurablcfunctionsandoffunctimishavingthepropertyofBaire,uniformconvergenceonsubsets

9．MetricandTopologicalSpaces

Definitions,completeandtopologicallycompletespaces,theBairecategorytheorem

10．ExamplesofMetricSpaces

Uniformandintegralmetricsinthespaceofcontinuousfunctions,integrablfunctions,pseudmetricspaces，thespaceofmeasurablesets

11．NowhereDifferentiableFunctions

Banach’Sapplicationofthecategorymethod

12．TheTheoremofAlexandroff．

RemetrizationofaG．subset，topologicallycompletesubspaces

13．TransformingLinearSetsintoNullsets’．

Thespaceofautomorphismsofaninterval．effectofmonotonesubstitutionORiemannintegrability．nullsetsequivalenttosetsoffirstcategory

14．Fubini’STheorem

Measurabilityandmeasureofsectionsofplanemeasurablesets

15．TheKuratowski．UlamTheorem．，

SectionsofplanesetshavingthepropertyofBaire．productsets，reducibilitytFubinistheorembymeansofaproducttransformation

16．TheBanachCategoryTheorem．

Opensetsoffirstcategoryormeasurezero,MontgomerySlemma，thethecrumsofMarczewskiandSikorski．cardinalsofmeasurezerodecompositionintoanullsetandasetoffirstcategory

17．ThePoincar6RecurrenceTheorem

Measureandcategoryofthesetofpointsrecurrentunderanondissipativtransformation，applicationtodynamicalsystems

18．TransitiveTransformations

Existenceoftransitiveautomorphismsofthesquare,thecategorymethod

19．TheSierpinski.Erd6sDualityTheorem．

Similaritiesbetweentheclassesofsetsofmeasurezeroandoffirstcategory，thprincipieofduality

20．ExamplesofDuality．

PropertiesofLusinsetsandtheirduals，setsalmostinvariantundertransformationsthatpreservenullsetsorcategory

21．TheExtendedPrincipleofDuality．

Acounterexample．productmeasuresandproductspaces，thezero-onelawanditscategoryanalogue

22．CategoryMeasureSpaces

Spacesinwhichmeasureandcategoryagree，topologiesgeneratedbylowedensities,theLebesguedensitytopology

SupplementaryNotesandRemarks．

References

SupplementaryReferences．

Index

　　Inthisedition,asetofSupplementaryNotesandRemarkshasbeenaddedattheend,groupedaccordingtochapter.Someofthesecallattentiontosubsequentdevelopments,othersaddfurtherexplanationoradditionalremarks.Mostoftheremarksareaccompaniedbyabrieflyindicatedproof,whichissometimesdifferentfromtheonegiveninthereferencecited.Thelistofreferenceshasbeenexpandedtoincludemanyrecentcontributions,butitisstillnotintendedtobeexhaustive.