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Topic: Borel algebra

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	Aljabar Borel - Wikipédia
		In the case X is a metric space, the Borel algebra in the first sense may be described generatively as follows: First define for any collection A of subsets of X (that is, for any subset of the power set P(X) of 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.
	su.wikipedia.org /wiki/Aljabar_Borel (546 words)

	[No title]
		Boolean algebra +------------------------------------------------------------ A Boolean algebra is a set S with two binary operations + and * which are commutative monoids (S,+,0), (S,,1) and satisfy the two distributive laws (x(y+z)=xy + yz, x+(yz) =(x+y)(x+z) as well as the complementary laws x*x=1, y+y=0.
		Examples are the algebra of classes, where + is the union and * is the intersection or the algebra of propositions, for which + is and and * is or.
		Borel measure +------------------------------------------------------------ A Borel measure is a measure on the sigma-algebra of Borel sets.
	www.math.harvard.edu /~knill/sofia/data/measuretheory.txt (702 words)

	Fungsi kaukur - Wikipédia
		By convention, if T is some topological space, such as the space of real numbers R or the complex numbers C, then the Borel σ-algebra generated by the open sets on T is used, unless otherwise specified.
		If a function from one topological space to another is measurable with respect to the Borel σ-algebras on the two spaces, the function is also known as a Borel function.
		Continuous functions are Borel, however, not all Borel functions are continuous.
	su.wikipedia.org /wiki/Measurable_function (177 words)

	Publications of K P S Bhaskara Rao (Site not responding. Last check: 2007-11-06)
		Complementation in the lattice of Borel structures, with E. Grzeorek and H. Sarbadhikari, Coll.
		Lattice of subagebras of a Boolean algebra, with M.B. Rao, Czechoslovak Math.
		The dual of a refinement algebra, with R.M. Shortt, Lecture Notes in pure and applied mathematics, General Topology and its applications, 5th North East Conference (Marcel Dekker), Vol.
	www.sckans.edu /~brao/pub.htm (1170 words)

	Borel algebra - InfoSearchPoint.com (Site not responding. Last check: 2007-11-06)
		The existence and uniqueness of the Borel algebra is shown by noting that the intersection of all σ-algebras containing T is itself a σ-algebra, so this intersection is the Borel algebra.
		The Borel algebra may alternatively and equivalently defined as the smallest σ-algebra which contains all the closed subsets of X.
		A particularly important example is the Borel algebra on the set of real numbers.
	www.infosearchpoint.com /display/Borel_set (194 words)

	Borel measure - Wikipedia, the free encyclopedia
		In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b − a (where a < b).
		The Borel measure is not complete, which is why in practice the complete Lebesgue measure is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree.
		(the Borel σ-algebra on E) is Borel iff
	en.wikipedia.org /wiki/Borel_measure (132 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Note that f[C_n] = range (f) intersect { y : y(n)=1} so f is an isomorphism between the measurable space with sigma-algebra generated by C and the subspace of the Borel algebra on Cantor space restricted to the range of f.
		I want a collection C where f is onto and some element of C is not Borel (actually, you should only need that the range of f is Borel for the following argument).
		The identity function viewed as a map from X_2 to X_1 is therefore Borel and injective, so the image of every Borel set should be Borel, which would imply that the algebra for X_1 should contain that for X_2.
	www.math.pitt.edu /~gartside/jc.txt (540 words)

	Wilmott Forums - Sigma-field question - very elementary (Site not responding. Last check: 2007-11-06)
		Is this a valid constituent of a Borel Algebra on R^3 : ((1,5), {5}, {2,4,6,8,10...
		For what its worth, its actually somewhat difficult to construct sets that aren't borel measurable (in the borel sigma algebra on the reals.) An example is if you break the reals into equivalence classes where two numbers are equivalent if they differ by a rational number.
		There are Borel measurable sets, and analytic sets, and Lebesgue measurable sets, however, you are probably not going to construct a pathological example by accident.
	www.wilmott.com /messageview.cfm?catid=8&threadid=31240 (1642 words)

	[No title]
		The intersection of all $\sigma$-algebras (algebras) containing $\cal J$ is the {\em minimal} $\sigma$-algebra (resp., algebra) containing $\cal J$.
		It is called the $\sigma$-algebra (resp., algebra) {\bf generated by} $\cal J$ and denoted by ${\cal B}({\cal J})$ (resp., by ${\cal A}({\cal J})$).\\ A simple but useful fact: if $\cal J$ is finite (countable), then ${\cal A}({\cal J})$ is also finite (countable).\\ \noindent{\sc 1.18 Definition}.
		We note how $T$ acts on the sequence of digits $i_0i_1i_2\ldots$ in both examples 2.27 and 2.28: the first (leftmost) digit is dropped and the rest of the sequence is moved (shifted) to the left, so that the second digit becomes the first, etc. We will see more of shift maps in Section~10.
	www.math.uab.edu /chernov/teaching/760notes (6141 words)

	Vignettes on automorphic and modular forms, representations, L-functions, and number theory
		Setting up Lie algebra action of sl(2) on Schwartz functions: archimedean case of Weil representation.
		The benefits of looking at the Lie algebra rather than Lie group action are compelling in this example.
		This implies that rho determines the isomorphism class of the (g,K)-module, and also that the whole representation space is the tensor product of the enveloping algebra U(p+) with rho.
	www.math.umn.edu /~garrett/m/v (1117 words)

	math lessons - Probability theory
		For an algebraic alternative to Kolmogorov's approach, see algebra of random variables.
		Some statisticians will assign probabilities only to events that they think of as random, according to their relative frequencies of occurrence, or to subsets of populations as proportions of the whole; those are frequentists.
		algebra arithmetic calculus equations geometry differential equations trigonometry number theory probability theory applied mathematics mathematical games mathematicians
	www.mathdaily.com /lessons/Probability_theory (754 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		In a probability context the sets are called events, and we use the interpretation P(F) = "the probability that the event F occurs".
		An important EMBED "Equation" \* mergeformat -algebra used to define random variables is the Borel EMBED "Equation" \* mergeformat -algebra on EMBED "Equation" \* mergeformat where EMBED "Equation" \* mergeformat is the real line.
		It can be shown that the Borel EMBED "Equation" \* mergeformat -algebra EMBED "Equation" \* mergeformat does indeed exist.
	www.unc.edu /~mserre/teaching/spring2004/envr275/lectureNotes/08_MeasureTheoryUsefulForStat.doc (258 words)

	Wilmott Forums - Souslin Space, EMH, and Derivative Pricing Model
		Some examples of Polish space are; (i) Rd with it usual topology, (ii) separable Banach space with it norm topology, (iii) compact Hausdorff space with countable basis, and (iv) the space [0, 1]N consisting of all sequences of zeros and ones.
		The Borel a-algebra of Souslin is countably generated.
		Roughly speaking, the existence of a disintegration of measure means that agents actions (beliefs) depend measurably on their information or that they put a full probability measure on the distinguishable events (if the Borel a-algebra is countably separated or generated).
	www.wilmott.com /messageview.cfm?catid=4&threadid=9886 (2547 words)

	[No title]
		Subject: Re: Measurability of metric Date: Sat, 16 Oct 1999 13:23:26 -0400 Newsgroups: sci.math Keywords: Borel sigma algebra on ExE is the square of that on E? In article
		The metric topology is the whole cardinal set 2^E The borel sigma-algebra is B(E)=2^E The product topology is the cardinal set 2^ExE So B(ExE)=2^ExE B(E)xB(E) is generated by rectangles AxB (E,d) is not seperable (as a metric space), but can be seperated by a countable set of subsets.
		I initially tried to prove that the diagonal was not in B(E)xB(E), but I am no longer sure whether this is true or not.
	www.math.niu.edu /~rusin/known-math/99/sigalg_prod (735 words)

	28: Measure and integration
		The Borel sets and related families are constructed as a part of "descriptive" set theory (now in section 03E).
		Chaotic attractors are treated in 37: Dynamical Systems; these may lead to fractal sets.
		When is the Borel sigma algebra on E x E the square of that on E? Regular signed Borel measures
	www.math.niu.edu /~rusin/known-math/index/28-XX.html (758 words)

	Measures and -algebras
		A Borel measure is said to be outer regular on
		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.
	www-math.mit.edu /~rbm/18.155-F02/Lecture-notes/node4.html (479 words)

	Borel measure
		In mathematics, the Borel algebra is the smallest σ-algebra on the real numbers R containing the intervals, and the Borel measure is the measure on this σ-algebra which gives to the interval [a, b] the measure b - a (where a < b).
		The Borel measure isn't complete, which is why in practice the complete Lebesgue measure is preferred: every Borel measurable set is also Lebesgue measurable, and the measures of the set agree.
		Take this note to be able to access this article instantly from any page
	www.fastload.org /bo/Borel_measure.html (127 words)

	List of mathematical topics (A-C) - Gurupedia (Site not responding. Last check: 2007-11-06)
		Algebraic closure -- Algebraic curve -- Algebraic element -- Algebraic enumeration -- Algebraic extension -- Algebraic form -- Algebraic geometry --
		Algebraic geometry and analytic geometry -- Algebraic group -- Algebraic integer -- Algebraic K-theory -- Algebraic number -- Algebraic number theory -- Algebraic surface -- Algebraic structure -- Algebraic topology --
		Algebraically independent -- Algorism -- Algorithm -- Algorithmic information theory -- Algorithmics -- Algorithms for calculating variance -- Aliasing -- Alignments of random points -- Aliquot sequence -- Aliquot stringing -- Al-Khwarizmi -- Almost all -- Almost complex manifold --
	www.gurupedia.com /l/li/list_of_mathematical_topics_(a-c).htm (2331 words)

	Some Ergodic Theory
		(Remember that a measure is a map from a certain collection (an algebra) of subsets
		In this case the algebra is the Borel
		In order to see that one has such a measure, first notice that Lebesgue measure is invariant for the map
	www.maths.warwick.ac.uk /~strien/MA424/HTMLversion/node14.html (103 words)

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