测度与范畴学

《测度与范畴学(第2版)》讲述了：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 目录 -------------------------------------------------------------------------------- 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