《測度與範疇學(第2版)》講述瞭：This book has two main themes: the Baire category theorem as a method for proving existence, and the "duality" between measure and category. The category method is illustrated by a variety of typical applications, and the analogy between measure and category is explored in all of its ramifications. To this end, the elements of metric topology are reviewed and the principal properties of Lebesgue measure are derived. It turns out that Lebesgue integration is not essential for present purposes——the Riemann integral is sufficient. Concepts of general measure theory and topology are introduced, but not just for the sake of generality. Needless to say, the term "category" refers always to Baire category; it has nothing to do with the term as it is used in homological algebra

目錄

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1.MeasureandCategoryontheLine

2.LiouvilleNumbers

3.LebesgueMeasureinr-Space

4.ThePropertyofBaire

5.Non-MeasurableSets

6.TheBanach-MazurGame

7.FunctionsofFirstClass

8.TheTheoremsofLusinandEgoroff

9.MetricandTopologicalSpaces

10.ExamplesofMetricSpaces

11.NowhereDifferentiableFunctions

12.TheTheoremofAlexandroff

13.TransformingLinearSetsintoNullsets

14.Fubini'sTheorem

15.TheKuratowski-UlamTheorem

16.TheBanachCategoryTheorem

17.ThePoincareRecurrenceTheorem

18.TransitiveTransformations

19.TheSierpinski-ErdosDualityTheorem

20.ExamplesofDuality

21.TheExtendedPrincipleofDuality

22.CategoryMeasureSpaces

SupplementaryNotesandRemarks

References

SupplementaryReferences

Index