A Course on Borel SetsSpringer, 1 dic 2013 - 261 pagine The roots of Borel sets go back to the work of Baire [8]. He was trying to come to grips with the abstract notion of a function introduced by Dirich let and Riemann. According to them, a function was to be an arbitrary correspondence between objects without giving any method or procedure by which the correspondence could be established. Since all the specific functions that one studied were determined by simple analytic expressions, Baire delineated those functions that can be constructed starting from con tinuous functions and iterating the operation 0/ pointwise limit on a se quence 0/ functions. These functions are now known as Baire functions. Lebesgue [65] and Borel [19] continued this work. In [19], Borel sets were defined for the first time. In his paper, Lebesgue made a systematic study of Baire functions and introduced many tools and techniques that are used even today. Among other results, he showed that Borel functions coincide with Baire functions. The study of Borel sets got an impetus from an error in Lebesgue's paper, which was spotted by Souslin. Lebesgue was trying to prove the following: Suppose / : )R2 -- R is a Baire function such that for every x, the equation /(x,y) = 0 has a. unique solution. Then y as a function 0/ x defined by the above equation is Baire. |
Sommario
| 1 | |
Standard Borel Spaces | 81 |
4 | 95 |
Analytic and Coanalytic Sets | 126 |
Selection and Uniformization Theorems | 183 |
4 | 195 |
xi | 213 |
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A-measurable A₁ admits a Borel ambiguous class analytic subsets assume B₁ B₂ Baire property Baire space Borel function Borel isomorphism Borel map Borel measurable Borel set Borel subset called Cantor cardinality Clearly closed set closed subset closed under countable coanalytic sets comeager completely metrizable continuous map contradiction convergent Corollary countable base countable intersections countable unions defined denote dense disjoint analytic equivalence classes Example Exercise finite function ƒ Gs set Hence homeomorphic induction irr(A irr(n Lebesgue measurable Lemma Let G lim sup map ƒ meager measurable space multifunction nonmeager Note o-algebra one-to-one open set ordinal pairwise disjoint partition pointclass Polish group Polish space Polish topology Proposition prove real numbers result follows second countable selection theorem sequence Souslin operation standard Borel space strong measure zero subset of NN Suppose Take topological space uncountable Polish space w₁ well-ordered set