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Topic: Nonmeasurable

	Countrybookshop.co.uk - Nonmeasurable Sets and Functions (Site not responding. Last check: 2007-11-03)
		The role of nonmeasurable sets (functions) in point set theory and real analysis is underlined and various classes of such sets (functions) are investigated.
		It is also demonstrated that there are close relationships between the existence of nonmeasurable sets and some deep questions of axiomatic set theory, infinite combinatorics, set-theoretical topology, general theory of commutative groups.
		Nonmeasurable sets associated with filters.; Appendix 1: Logical aspects of the existence of nonmeasurable sets.
	www.countrybookshop.co.uk /books/index.phtml?whatfor=0444516263 (245 words)

	Axiom of dependent choice - Wikipedia, the free encyclopedia
		In mathematics, the axiom of dependent choice, denoted DC, is a weak form of the axiom of choice which is still sufficient to develop most of real analysis.
		Unlike the full axiom of choice (AC), DC is insufficient to prove (given ZF) that there is a nonmeasurable set of reals, or that there is a set of reals without the property of Baire or without the perfect set property.
		The axiom can be stated as follows: For any nonempty set X and any entire binary relation R on X, there is a sequence (x
	en.wikipedia.org /wiki/Axiom_of_dependent_choice (202 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Inverse-scattering method for dielectric objects based on the reconstruction of the nonmeasurable equivalent current density
		The approach is based on the reconstruction of the radiating (or, in general, the measurable) components of the equivalent current density, by means of a singular value decomposition of the discretized Green operator.
		Such components are then inserted in a nonlinear equation whose unknowns are the nonmeasurable components as well as the dielectric features of the body under test.
	www.agu.org /pubs/abs/rs/1998RS900009/1998RS900009.html (297 words)

	Ars Mathematica » Blog Archive » The myth of measurable sets
		The one example of a nonmeasurable set anyone sees is that of Vitali, which requires the axiom of choice, so it’s tempting to believe that without the axiom of choice, every set is measurable.
		This belief is only reenforced by a result of Solovay’s that the axioms of set theory, removing the axiom of choice, and adding the axiom that every set is measurable remain consistent.
		The sets are not exactly nonmeasurable, but their measurability is independent of ZFC.
	www.arsmathematica.net /archives/2005/09/21/the-myth-of-measurable-sets (300 words)

	Allegheny College: Academics: Mathematics Department:2000 class (Site not responding. Last check: 2007-11-03)
		The desirable properties of an extension of length are discussed and several are proved to hold for Lebesgue outer measure.
		As well, a charactorization is given of those sets, for which all of the desirable properties hold.
		A construction of the Vitali nonmeasurable set is given as well as a proof that the set is indeed nonmeasurable.
	webpub.alleg.edu /dept/mathweb/2000Abstracts.htm (788 words)

	The Mathematica Journal: Volume 3, Issue 4 (Site not responding. Last check: 2007-11-03)
		The Banach-Tarski Paradox asserts that a solid ball in 3-space may be decomposed into five disjoint sets that can be rearranged to form two solid balls, each the same size as the original ball.
		The sets are nonmeasurable, so it is impossible to visualize the paradox.
		However, the algebraic idea underlying the paradox can be given a constructive interpretation in the hyperbolic plane.
	www.mathematica-journal.com /issue/v3i4/article/wagon (109 words)

	[No title]
		Moreover, your first sentence is false under any "explicit" notion I know of, but I believe that the Axiom of Choice (even for just alef_1 many collections of sets of reals) ISN'T a consequence of the existence of a nonmeasurable set.
		If you allow me to make use of ONE nonmeasurable set, then I can give you uncountably many others in a simple way.
		When I first typed up my response, I was thinking one of the questions had to do with an explicit presentation of uncountably many nonmeasurable sets.
	www.math.niu.edu /~rusin/known-math/97/measure (1034 words)

	Georgian Math. J. (Site not responding. Last check: 2007-11-03)
		We prove that there exist two absolutely negligible subsets $A$ and $B$ of the real line $R$, whose algebraic sum $A + B$ is an absolutely nonmeasurable subset of $R$.
		We also obtain some generalization of this result and formulate a relative open problem for uncountable commutative groups.
		Invariant measure, quasi-invariant measure, absolutely negligible set, absolutely nonmeasurable set, extension of measure.
	www.emis.de /journals/GMJ/vol12/12-3-7.htm (75 words)

	Atlas: Complete nonmeasurabilitty in regular families of small sets by Szymon Zeberski
		Namely, the intersection of this union with any measurable set that is not in I is nonmeasurable (recall, the measurability is understood here in the sense of belonging to the
		In this paper we show how to obtain complete nonmeasurability of the union of subfamily of A assuming that the family A is in some sense regular.
		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 # catc-07.
	atlas-conferences.com /cgi-bin/abstract/catc-07 (425 words)

	Clinical Trials Research Unit, Current Cancer Trials, Leukemia (Site not responding. Last check: 2007-11-03)
		Patients must have histologically or cytologically confirmed adenocarcinoma of the breast with measurable or nonmeasurable locally recurrent or metastatic disease.
		All scans or x-rays used to document measurable or nonmeasurable disease must be done within 4 weeks prior to randomization.
		Patients with breast cancer overexpressing HER-2 (gene amplification by FISH or 3+ overexpresslon by immunohistochemistry), are not eligible unless they have received prior therapy with Herceptin.
	www.hsc.wvu.edu /mbrcc/ctru/cancer/breast/ecog2100.htm (849 words)

	Georgian Math. J. (Site not responding. Last check: 2007-11-03)
		The notions of a negligible set and of an absolutely nonmeasurable set are introduced and discussed in connection with the measure extension problem.
		of the plane ${\bf R}^2$ which are $T_2$-negligible and, simultaneously, $G$-absolutely nonmeasurable.
		Sierpinski partition, quasi-invariant measure, uniform set, negligible set, absolutely nonmeasurable set.
	www.univie.ac.at /EMIS/journals/GMJ/vol11/11-3-7.htm (107 words)

	Mathematics Magazine: April 2003
		The cube of a number is exhibited as an arithmetic progression that begins with a triangular number.
		We construct a nonmeasurable subset of the torus, similar to the well-known Vitali subset of the unit interval.
		Our set is "visual" in the sense that we can draw the elements used in constructing the set.
	www.maa.org /pubs/mag_apr03_toc.html (1000 words)

	IB-OR-2000-112-Attachment 3 (Site not responding. Last check: 2007-11-03)
		The spreadsheet will also identify (1) the types of projects undertaken by local Forest Service and BLM units; (2) the program goals met by the projects undertaken; and (3) whether measurable or nonmeasurable savings and other benefits have resulted from the projects undertaken.
		Thus, in some instances, the savings and benefits may be both measurable and nonmeasurable at the same time and the appropriate cells completed.
		In those instances where dollar savings are not achieved but certain costs were avoided because of the shared initiative, include this amount in the cost avoidance cell and provide GAO with your calculations of the amount.
	www.blm.gov /nhp/efoia/or/fy2000/ib/b2000-112att3.htm (1592 words)

	Biweekly 72-Hour 9-Aminocamptothecin Infusion As Second-Line Therapy for Ovarian Carcinoma: Phase II Study of the New ...
		their disease was measurable, or nonmeasurable but assessable.
		nonmeasurable but assessable disease during a period of 4 weeks
		28 in the nonmeasurable but assessable disease arm.
	www.jco.org /cgi/content/full/22/1/120 (3547 words)

	FAPM LETTER 430-4 AVR PERFORMANCE PLAN (Site not responding. Last check: 2007-11-03)
		Recognition having significant monetary value may be conferred for both measurable and nonmeasurable benefits.
		The scales in the next section are intended to provide a common understanding of the appropriate level of reward for various types of contributions.
		Criteria for determining the value of a nonmeasurable benefits contribution worthy of recognition should include, but not be limited to:
	www.faa.gov /ahr/policy/fapm/fapms/fapm4304.cfm (6428 words)

	Convergence in distribution of nonmeasurable random elements, Patrizia Berti, Pietro Rigo
		Convergence in distribution of nonmeasurable random elements, Patrizia Berti, Pietro Rigo
		Further, it is shown that the empirical process for an exchangeable sequence can fail to converge, due to the nonexistence of any measurable limit, although it converges for an i.i.d.
		Because of phenomena of this type, Hoffmann-Jørgensen's definition is extended to the case of a nonmeasurable limit.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.aop/1078415839 (472 words)

	OUP: UK General Catalogue
		The author shows how this is related to famous problems by Lebesgue, Banach, and Ulam on the existence of certain measures on abstract sets, with corresponding algebras being algebras of measurable subsets with respect to these measures.
		In particular it is shown that for a certain algebra not to coincide with the algebra of all subsets of $X$ is equivalent to the existence of a nonmeasurable set with respect to a given measure.
		Although these questions don't seem to be related to mathematical logic, many results in this area were proved by ``metamathematical'' methods, using the method of forcing and other tools related to axiomatic set theory.
	www.oup.com /uk/catalogue/?ci=9780821827659 (382 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Given AC, of course this is the same as the cardinality of the continuum, but in models of ZF+AD it's bigger.
		> Anyway, as far as I know (which isn't > saying much), constructions of nonmeasurable sets use more than > continuum many choices, which is why I asked.
		Does the axiom of choice > for all families of continuum many nonempty sets imply the existence > of nonmeasurable sets?
	www.math.niu.edu /~rusin/known-math/01_incoming/E_0 (169 words)

	Nonmeasurable functions (Site not responding. Last check: 2007-11-03)
		I used this handout in a first year graduate course in real variables.
		The idea was to to do a little more with nonmeasurable sets than we found in our textbook.
		The notes end up interpreting the usual construction of a nonmeasurable set as providing a "paradoxical decomposition" of the unit circle.
	www.mth.msu.edu /~shapiro/Pubvit/Downloads/Nonmeas/Nonmsble.html (75 words)

	Journal of Applied Analysis, Vol. 2, No. 1, pp. 41-48, 1996 (Site not responding. Last check: 2007-11-03)
		Abstract: Starting with the classical Sierpi\'nski partition [7], we consider a question concerning the existence of a nonmeasurable set in the product measure space.
		Also, we discuss an analogous question on the existence of a set without the Baire property in the product topological space.
		In particular, it is shown that the situations of measure and category are essentially different.
	www.maths.tcd.ie /EMIS/journals/JAA/vol2i1/3.html (91 words)

	Abstracts of Joseph Y. Halpern's Publications
		In particular, we show here how to extend the definition of conditional probability to nonmeasurable sets, in order to get notions of inner and outer conditional probabilities, which can be viewed as best approximations to the true conditional probability, given our lack of information.
		As we show in a companion paper, the general (nonmeasurable) case corresponds precisely to replacing probability measures by Dempster-Shafer belief functions.
		For a nonmeasurable event (one to which we do not assign a probability), we can talk about only the inner measure and outer measure of the event.
	www.cs.cornell.edu /home/halpern/abstract.html (17995 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		() On nonmeasurable* subgroups of the real line, JournalO of Applied Analysis, v.1, n.2, 1995.
		() On the existance of nonmeasurable* subgroups of commutative groups, Real Analisys Exchange, v.27, n.1, 2001-2002.
		Journal, v.11, n.4, 2004 () Nonmeasurable* sets and functions,Elsevier Amsterdam, 2004.
	www.tech.org.ge /science_department/umaglesi_skola/1/01/1.01.90/index.doc (1331 words)

	Publisher description for Library of Congress control number 2004049706 (Site not responding. Last check: 2007-11-03)
		Publisher description for Nonmeasurable sets and functions / A.B. Kharazishvili.
		• highlights the importance of nonmeasurable sets (functions) for general measure extension problem.
		• Each chapter ends with exercises which provide valuable additional information about nonmeasurable sets and functions.
	www.loc.gov /catdir/description/els051/2004049706.html (258 words)

	Amazon.ca: Nonmeasurable Sets and Functions: Books (Site not responding. Last check: 2007-11-03)
		Be the first person to review this item.
		Look for books like Nonmeasurable Sets and Functions by subject:
		Top of Page : Nonmeasurable Sets and Functions
	www.amazon.ca /exec/obidos/ASIN/0444516263 (266 words)

	Journal of Applied Analysis, Vol. 2, No. 2, pp. 171-181, 1996 (Site not responding. Last check: 2007-11-03)
		Abstract: We prove that, for every nonzero $\sigma$-finite measure $\mu$ defined on the real line $R$ and invariant (or quasiinvariant) under all translations of $R$, there exists a subgroup of $R$ nonmeasurable with respect to $\mu$.
		Some generalizations of this result are discussed, too, and several problems related to them are posed.
		Keywords: Real line, invariant measure, quasiinvariant measure, nonmeasurable subgroup, Hamel basis, Ulam matrix,uncountable commutative group, Jonsson group
	www.univie.ac.at /EMIS/journals/JAA/vol2i2/3.html (91 words)

	HMC Math 132: Mathematical Analysis II, Fall 2005 (Site not responding. Last check: 2007-11-03)
		a) Let P, subset of [0,1), be the nonmeasurable set discussed in class.
		Show that if E is a measurable subset of P then m(E) = 0.
		c) Show that every set A of positive measure contains a nonmeasurable set.
	www.math.hmc.edu /~castro/HWMath132-Fall05.html (405 words)

	Legal Theory Blog (Site not responding. Last check: 2007-11-03)
		Indeed, as explained in Part III of the Article, federal regulation is more likely to be necessary to maximize welfare in the context of interstate losses in existence value than in the context of interstate physical effects, such as air or water pollution crossing state lines.
		The principal claim of those who reject the use of existence values as a rationale for federal regulation is that existence values are nonmeasurable and hence unsuitable for consideration in public policy.
		As explored in Part IV of the Article, this empirical objection is inconsistent with the findings of contingent value (CV) surveys in which respondents have been asked how much they would be willing to pay for the preservation of one or more natural resources.
	lsolum.blogspot.com /archives/2003_12_01_lsolum_archive.html (12304 words)

	5.1.3 Basic Measure Theory Definitions
		Therefore, it is hard to visualize the problem; see [
		838] for a construction of the bizarre nonmeasurable sets.
		By using Borel sets, the nastiness of nonmeasurable sets is safely avoided.
	msl.cs.uiuc.edu /planning/node190.html (546 words)

	Wiley::Acoustic Echo and Noise Control: A Practical Approach
		Application of these algorithms to acoustic echo cancellation and residual echo and noise suppression
		Estimation of nonmeasurable quantities necessary to control algorithms
		Requiring only a basic knowledge of linear systems theory and digital signal processing, this text offers an ideal resource for students, researchers, and systems engineers seeking to familiarize themselves with the latest developments and practical applications in this fascinating field.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471453463.html (353 words)

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