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	[No title]
		Borel measure +------------------------------------------------------------ A Borel measure is a measure on the sigma-algebra of Borel sets.
		Borel set +------------------------------------------------------------ A Borel set (=Borel measurable set) in a topological space is an element in the smallest sigma-algebra which contains all compact sets.
		Borel set +------------------------------------------------------------ The smallest sigma-algebra A of subsets of a topological space (X,O) containing O is called a Borel sigma-algebra.
	www.math.harvard.edu /~knill/sofia/data/measuretheory.txt (702 words)

	Lebesgue measure - Wikipedia, the free encyclopedia
		Sets which can be assigned a volume are called Lebesgue measurable; the volume or measure of the Lebesgue measurable set A is denoted by λ(A).
		If A is a disjoint union of finitely many or countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ(A) is equal to the sum (or infinite series) of the measures of the involved measurable sets.
		The Borel measure agrees with the Lebesgue measure on those sets for which it is defined; however, there are many more Lebesgue-measurable sets than there are Borel measurable sets.
	en.wikipedia.org /wiki/Lebesgue_measure (615 words)

	Borel algebra -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-11)
		Here, the minimal σ-algebra containing a collection T of (A set whose members are members of another set; a set contained within another set) subsets of X is the smallest σ-algebra containing T.
		In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows.
		The Borel algebra on the reals is the smallest σ-algebra on R which contains all the (The difference in pitch between two notes) intervals.
	www.absoluteastronomy.com /encyclopedia/b/bo/borel_algebra.htm (765 words)

	Borel algebra - LearnThis.Info Enclyclopedia (Site not responding. Last check: 2007-09-11)
		In mathematics, the Borel algebra (or Borel σ-algebra) on a topological space is either of two σ-algebras;s on a topological space X:
		It is the algebra on which the Borel measure is defined.
		The Borel algebra on the reals is the smallest sigma algebra on R which contains all the intervals.
	encyclopedia.learnthis.info /b/bo/borel_algebra.html (535 words)

	Haar measure - Wikipedia, the free encyclopedia
		In this article, the σ-algebra generated by all compact subsets of G is called the Borel algebra.
		An element of the Borel algebra is called a Borel set.
		Unless G is a discrete group, it is impossible to define a countably-additive right invariant measure on all subsets of G, assuming the axiom of choice.
	en.wikipedia.org /wiki/Haar_measure (784 words)

	Search Results for Borel
		Borel, although not the first to define the sum of a divergent series, was the first to develop a systematic theory for a divergent series which he did in 1899.
		This result was strengthened by Borel in 1899 when he proved a lower bound for P(e), where P is a polynomial with integer coefficients, depending on the maximum modulus of the integer coefficients of P. Gelfond, Feldman's supervisor, had extended Borel's result to numbers of the form alpha beta, where alpha, beta are algebraic numbers.
		Feldman proved in his thesis Borel type results (called the measure of transcendence) for logarithms of algebraic numbers, obtaining estimates for the lower bound depending (as did Gelfond) on both the degree of P and the maximum modulus of its coefficients.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Borel&CONTEXT=1 (1912 words)

	Borel algebra Article, Borelalgebra Information (Site not responding. Last check: 2007-09-11)
		In mathematics, the Borel algebra (or Borelσ-algebra) on a topological space is either of two σ-algebras on a topological space X:
		It should be noted that as Borel spaces R and R union with a countable set, areisomorphic.
		For subsets of Polish spaces, Borel sets can be characterized as those sets which are the ranges of continuous injective mapsdefined on Polish spaces.
	www.anoca.org /space/set/borel_algebra.html (530 words)

	[No title] (Site not responding. Last check: 2007-09-11)
		Date: Tue, 15 Oct 1996 12:14:08 -0500 Jonathan King wrote: > > On the unit interval, the field (sigma-algebra) B of Borel sets (the smallest > field containing the open sets) is a proper subfield of L, the Lebesgue > measurable sets.
		The measure of K is 0 so every subset of K is Lebesgue measurable, but not every subset of K is Borel, since K has 2^c subsets and there are only c Borel sets.
		Now there exists a bijection from K to K that takes X to Y and Y to X; fix one such, extend it to a bijection of the unit interval to itself by declaring it to be constant on the complement of K, and call it f.
	www.math.niu.edu /~rusin/known-math/96/borel.sets (309 words)

	Olivier Finkel
		We prove that omega-CFL exhaust the hierarchy of Borel sets of finite rank, and that one cannot decide the borel class of an omega-CFL, giving an answer to a question of Lescow and Thomas [Logical Specifications of Infinite Computations, In:"A Decade of Concurrency", Springer LNCS 803 (1994), 583-621].
		We study the set C(F) of points of continuity of an omega rational function F which is always a Borel Pi -set.
		Conversely every Pi omega regular set is the set of points of continuity C(F) of some function F accepted by a synchronous 2-tape Büchi automaton.
	www.logique.jussieu.fr /www.finkel (3976 words)

	Math218
		Continuity of functions on a metric space, proof of the fact that a function is continuous iff the preimage of an open set is open.
		Let {f_n} be a sequence of measurable functions on R. Then the set of those values x for which the limit lim f_n(x) exists is measurable.
		Construct an increasing function on R whose set of discontinuities is Q (rationals).
	math.vanderbilt.edu /~neamtu/330a/topics.html (1558 words)

	[No title]
		We say that a class of sets satisfies CH (or the sets individually satisfy CH) if each of the sets has countable cardinality or cardinality c.
		One could argue that Cantor showed open sets satisfy CH in 1873-74 when he introduced cardinality, since it is easy to see that every open interval has the same cardinality as the reals.
		Souslin (1916, I think) showed that the CA sets are either countable, have cardinality aleph_1, or have cardinality c.
	www.math.niu.edu /~rusin/known-math/01_incoming/aleph1 (1073 words)

	ESI The Erwin Schrdinger International Institute for Mathematical Physics (Site not responding. Last check: 2007-09-11)
		In Descriptive Set Theory one tries to avoid these pathologies by concentrating on natural classes of well-behaved sets of reals, like Borel sets or projective sets (the smallest class of sets containing Borel sets and closed under projections from higher dimenional spaces).
		While this is a restricted class of sets it includes most of the sets that arise naturally in mathematical practice.
		Descriptive set theory provides a general framework and tools for studying these problems and it allows us to properly formulate and give precise answers to some previously vague classification problems in various areas of mathematics.
	www.esi.ac.at /Programs/SetTheory-WS04.html (474 words)

	[No title]
		A $\sigma${\bf -algebra} $\cal B$ for a set $X$ is a collection of subsets of $X$ such that \begin{itemize} \item[(i)] $\emptyset,X\in{\cal B}$ \item[(ii)] $\{B_i\}_{i=1}^{\infty}\in{\cal B}\ \ \Longrightarrow\ \ \cup_{i=1}^{\infty}B_i\in{\cal B}$ \item[(iii)] $B\in{\cal B}\ \ \Longrightarrow\ \ B^c\in{\cal B}$ \end{itemize} A pair $(X,{\cal B})$ is called a {\bf measurable space}.
		The intersection of any family of $\sigma$-algebras of a set $X$ is always a $\sigma$-algebra of $X$ (the family itself may be finite, countable or uncountable).
		Suppose the measure $\rho_{\mu}$ in 7.15 is concentrated on a finite or countable set of ergodic measures $\nu_1,\nu_2,\ldots$.
	www.math.uab.edu /chernov/teaching/760notes (6141 words)

	No Title (Site not responding. Last check: 2007-09-11)
		If h is the indicator of a Borel set, (2) follows by the definition of the Lebesgue integral.
		Since an arbitrary non-negative Borel function is a monotone limit of simple functions, by the monotone convergence theorem (2) extends to the case of non-negative functions.
		Finally, if h is a bounded Borel function, it can be expressed as a difference of two bounded non-negative functions, for both of which the expressions on the left and on the right of (2) are finite due to their boundedness.
	www-math.cudenver.edu /~puhalski/teaching/99-prob/t1r/t1r.html (316 words)

	Reverse mathematics
		Most relevant sets of real numbers, including all Borel sets, can be coded by real numbers with the membership relation expressible in second order arithmetic.
		The primary difference between doing classical mathematics in set theory (ZFC) and doing it in second order arithmetic is that in second order arithmetic one deals with codes for sets rather than sets themselves (except sets of integers).
		Under the correct formalizations, most of the general theorems are actually equivalent to the minimal canonical axiom required for their proof.
	www.brainyencyclopedia.com /encyclopedia/r/re/reverse_mathematics.html (288 words)

	Publications, Jacques Duparc (Site not responding. Last check: 2007-09-11)
		In more technical terms: we define simple Borel operations which are homomorphic to ordinal sum, to multiplication by a countable ordinal, and to ordinal exponentiation of base kappa, under the map which sends every Borel set A of finite rank to its Wadge degree.
		For each Wadge class of Borel sets of reals, as described in Louveau's works, we produce a set that is complete for this class, and is canonical in the sense it is defined by operations on sets that are the set theoretic counterparts of natural operations on ordinals.
		For each Borel set of reals A, of finite rank, we obtain a ''normal form'' of A, by finding a Borel set Omega of maximum simplicity, such that A and Omega continuously reduce to each other.
	www-mgi.informatik.rwth-aachen.de /Publications/Duparc/index-abstracts.html (1166 words)

	Citations: Borel sets and circuit complexity - Sipser (ResearchIndex) (Site not responding. Last check: 2007-09-11)
		It is not surprising that this carries over to the parameterized setting to show that the G[t] hierarchy is proper, but it is noteworthy that it extends to our nondeterministic classes: Theorem 4.1 For all t 1, N [t] ae N [t 1] Proof.
		Sipser, Borel sets and circuit complexity, in: Proceedings of the 15th ACM Symposium on Theory of Computing (1983) 61-69.
		Generic sets were introduced by Cohen [4] as a tool for proving independence results in set theory.
	citeseer.ist.psu.edu /context/156810/0 (2896 words)

	[No title]
		T is the set of all real numbers x with continued fraction expansion 1 x = a[0] + -------------------------- 1 a[1] + ------------------ 1 a[2] + ---------...
		Then any set of the form E union A, where A is a measurable subset of [2,3], is measurable and non-Borel.
		The Axiom of Determinateness implies that all sets are Lebesgue measurable.
	www.math.niu.edu /~rusin/known-math/97/measure (1034 words)

	Descriptive Set Theory. By K.Podnieks (Site not responding. Last check: 2007-09-11)
		In the descriptive set theory the meaning of "simple", "definable" sets (of real numbers) is defined explicitly by introducing the so-called Borel sets and projective sets.
		each PI set is obtained from a Borel set by applying, first, a continuous image operations, and then, by applying the complement operation.
		The class of projective sets is closed is closed under finite unions, finite intersections, and inverse continuous images, yet (unlike the class of Borel sets) it is not closed under countable unions and countable intersections.
	www.ltn.lv /~podnieks/gtaa.html (1770 words)

	Articles - Well-behaved (Site not responding. Last check: 2007-09-11)
		Mathematicians (and those in related sciences) very frequently speak of whether a mathematical object -- a number, a function, a set, a space of one sort or another -- is "well-behaved" or not.
		Continuous functions are better-behaved than Riemann-integrable functions on compact sets in calculus.
		Borel sets are better-behaved than arbitrary sets of real numbers.
	www.cbasket.com /articles/Well-behaved (304 words)

	More measure theory (Site not responding. Last check: 2007-09-11)
		Informal and imprecise remarks for 21.3 and 21.4: The calculations in these sections should show that any set which is "constructed via any reasonable algorithm" from measurable sets is measurable.
		To see that, consider any open set in the product topology; it can be written as a union of countably many open rectangles.
		This net of functions converges monotonely and dominatedly to the constant function 1, but the integrals are all 0, which does not converge to 1.
	math.vanderbilt.edu /~schectex/ccc/addenda/measure.html (2017 words)

	[No title]
		A set is measurable if and only if its characteristic function is a measurable map.
		Feb.25: The outer measure on R. The general definition of outer measure and Lebesgue measurable sets.
		Existence of Lebesgue measurable sets which are not Borel.
	www.math.uiuc.edu /~fboca/math441.html (470 words)

	AMCA: Borel Universal Sets of Compact Spaces by Joseph T H Lo (Site not responding. Last check: 2007-09-11)
		We shall investigate what properties of X can be inferred from properties of Y if X has such a Borel universal set parametrised by Y. Special attention will be given to the influence of compactness of X. We note in particular the following results.
		There is a non-compact space X with an open universal set parametrised by Y such that hd(X) > hd(Y), and there is a consistent example of a non-compact S-space with an open universal set parametrised by an L-space.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/d/g/06.htm (229 words)

	Measures and -algebras
		Thus the measure constructed in Proposition 2.5 is outer regular on all Borel sets!
		Here we need to know that compact sets are Borel measurable.
		Notice how similar this is to one of the characterizations of continuity for maps between metric spaces in terms of open sets.
	www-math.mit.edu /~rbm/18.155-F02/Lecture-notes/node4.html (479 words)

	List KWIC DDC22 510 and MSC+ZDM E-N lexical connection
		set (change of topology, comparison of topologies, lattices of topologies) # several topologies on one
		set functions and measures on topological spaces (regularity of measures, etc.)
		sets of sentences # undecidability and degrees of
	www.math.unipd.it /~biblio/kwic/msc-cdd/dml2_11_51.htm (1190 words)

	A normal form for Borel sets (Extended Abstract) (ResearchIndex) (Site not responding. Last check: 2007-09-11)
		A normal form for Borel sets (Extended Abstract)
		In case of Borel sets of finite rank, we prove the above result essentially by defining simple Borel...
		1.7: The normal form of Borel sets - Part II: Borel sets of infinite..
	citeseer.ist.psu.edu /483962.html (199 words)

	Math 512 Descriptive Set Theory (Site not responding. Last check: 2007-09-11)
		we try to avoid these pathologies by concentrating on natural classes of well-behaved sets of reals, like Borel sets or projective sets (the smallest class of sets containing Borel sets and closed under projections from higher dimenional spaces).
		Lately there have been many intersting connections with dynamical systems, through the study of orbit equivalence relations.
		Basics of topology of metric spaces, ideally students should be familiar with very basic set theory (cardinals and ordinals) and elementary measure theory, but these topics can be picked up as we go along.
	www.math.uic.edu /~marker/math512 (278 words)

	Atlas: Borel Universal Sets by Joseph T H Lo (Site not responding. Last check: 2007-09-11)
		Let \Gamma be a class function mapping a topological space X to a family \Gamma(X) of subsets of X. A subset U of X ×Y is a \Gamma-universal set for X parametrised by Y if U
		We will consider how the properties of the parametrising space Y affect those of the space X, in the case when \Gamma(X) is a Borel class of X. Example results include:
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-25.
	atlas-conferences.com /cgi-bin/abstract/cacl-25 (178 words)

	Math218 (Site not responding. Last check: 2007-09-11)
		Some basic principles of set theory (The Hausdorff maximal principle, Zorn's lemma, the well ordering principle, axiom of choice).
		The extended real number system, characterization of open sets in \RR.
		Metric spaces (definition, open and closed sets, closure and interior of sets, denseness, nowhere denseness, separability).
	math.vanderbilt.edu /~neamtu/330a/final.html (489 words)

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