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Статті в журналах з теми "Borel measure"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Zhang, Chun Qin, and Hui Zhang. "Borel-Cantelli Lemma for Sugeno Measure." Applied Mechanics and Materials 614 (September 2014): 367–70. http://dx.doi.org/10.4028/www.scientific.net/amm.614.367.

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Анотація:

Sugeno measure is a fuzzy measure. In this paper, we derive the Borel-Cantelli lemma for Sugeno measure. This result is a natural extension of the classical Borel-Cantelli lemma to the case where the measure tool is fuzzy. The properties of Sugeno measure are further discussed. Then the Borel-Cantelli lemma will be proven on Sugeno measure space. This work generalizes the research and applications of the Borel-Cantelli lemma.

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2

Aldaz, J. M., та H. Render. "Borel Measure Extensions of Measures Defined on Sub-σ-Algebras". Advances in Mathematics 150, № 2 (березень 2000): 233–63. http://dx.doi.org/10.1006/aima.1999.1866.

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3

Schilling, Kenneth. "Vanishing Borel sets." Journal of Symbolic Logic 63, no. 1 (March 1998): 262–68. http://dx.doi.org/10.2307/2586600.

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Анотація:

Henson and Ross [1] answered the question of when two hyperfinite sets A, B in an ℵ1-saturated nonstandard universe are bijective by a Borel function: precisely when ∣A∣/∣B∣ ≈ 1. Živaljević [5] generalized this result to nonvanishing Borel sets. He defined a set to be nonvanishing if it is Loeb-measurable and has finite, non-zero measure with respect to some Loeb counting measure. He then showed that two nonvanishing Borel sets are Borel bijective just in case they have the same finite, non-zero measure with respect to some Loeb counting measure.Here we shall complete the cycle, for Borel sets at least. For N ∈ *N, let λN be the internal counting measure given by λN(A) = ∣A∣/N for A internal. Then for vanishing Loeb-measurable sets B, it is natural to consider the Dedekind cut (BL, BR) on *N consisting of those N for which B has 0λN-measure infinity and zero, respectively. We show that, for all vanishing Borel sets B, B and BL are Borel bijective. It follows that vanishing Borel sets B and C are Borel bijective if, and only if, BL = CL. Combined with Živaljević's result, we can characterize when arbitrary Borel sets are Borel bijective: precisely when they have the same measure with respect to all Loeb counting measures.In the final section, we generalize in a similar way results of [2] and [5] to characterize when two Borel sets are bijective by a countably determined function: precisely when, for all N, one has 0λN-measure 0 if and only if the other also has 0λN-measure 0.

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4

Dostál, Petr, Jaroslav Lukeš, and Jiří Spurný. "Measure Convex and Measure Extremal Sets." Canadian Mathematical Bulletin 49, no. 4 (December 1, 2006): 536–48. http://dx.doi.org/10.4153/cmb-2006-051-1.

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Анотація:

AbstractWe prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

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5

INSELMANN, MANUEL J., and BENJAMIN D. MILLER. "RECURRENCE AND THE EXISTENCE OF INVARIANT MEASURES." Journal of Symbolic Logic 86, no. 1 (February 15, 2021): 60–76. http://dx.doi.org/10.1017/jsl.2020.8.

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AbstractWe show that recurrence conditions do not yield invariant Borel probability measures in the descriptive set-theoretic milieu, in the strong sense that if a Borel action of a locally compact Polish group on a standard Borel space satisfies such a condition but does not have an orbit supporting an invariant Borel probability measure, then there is an invariant Borel set on which the action satisfies the condition but does not have an invariant Borel probability measure.

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6

Panchapagesan, T. V., and Shivappa Veerappa Palled. "On vector lattice-valued measures II." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 40, no. 2 (April 1986): 234–52. http://dx.doi.org/10.1017/s144678870002721x.

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AbstractFor a weakly (, )-distributive vector lattice V, it is proved that a V {}-valued Baire measure 0 on a locally compact Hausdorff space T admits uniquely regular Borel and weakly Borel extensions on T if and only if 0 is strongly regular at . Consequently, for such a vector lattice V every V-valued Baire measure on a locally compact Hausdorff space T has unique regular Borel and weakly Borel extensions. Finally some characterisations of a weakly (, )-distributive vector lattice are given in terms of the existence of regular Borel (weakly Borel) extensions of certain V {}-valued Barie measures on locally compact Hausdorff spaces.

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7

Elekes, Márton, та Tamás Keleti. "Is Lebesgue measure the only σ-finite invariant Borel measure?" Journal of Mathematical Analysis and Applications 321, № 1 (вересень 2006): 445–51. http://dx.doi.org/10.1016/j.jmaa.2005.08.035.

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8

Grubb, D. J., and Tim LaBerge. "Spaces of Quasi-Measures." Canadian Mathematical Bulletin 42, no. 3 (September 1, 1999): 291–97. http://dx.doi.org/10.4153/cmb-1999-035-5.

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AbstractWe give a direct proof that the space of Baire quasi-measures on a completely regular space (or the space of Borel quasi-measures on a normal space) is compact Hausdorff. We show that it is possible for the space of Borel quasi-measures on a non-normal space to be non-compact. This result also provides an example of a Baire quasi-measure that has no extension to a Borel quasi-measure. Finally, we give a concise proof of theWheeler-Shakmatov theorem, which states that if X is normal and dim(X) ≤ 1, then every quasi-measure on X extends to a measure.

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9

Conley, Clinton, and Benjamin Miller. "Measure reducibility of countable Borel equivalence relations." Annals of Mathematics 185, no. 2 (March 1, 2017): 347–402. http://dx.doi.org/10.4007/annals.2017.185.2.1.

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10

Eigen, S., A. Hajian, and B. Weiss. "Borel automorphisms with no finite invariant measure." Proceedings of the American Mathematical Society 126, no. 12 (1998): 3619–23. http://dx.doi.org/10.1090/s0002-9939-98-04489-x.

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11

Balcerzak, M., and S. Gła̧b. "Measure-category properties of Borel plane sets and Borel functions of two variables." Acta Mathematica Hungarica 126, no. 3 (December 18, 2009): 241–52. http://dx.doi.org/10.1007/s10474-009-9055-4.

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12

Olsen, L. "Multifractal dimensions of product measures." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 4 (November 1996): 709–34. http://dx.doi.org/10.1017/s0305004100001675.

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Анотація:

AbstractWe study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

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13

MILLER, BENJAMIN. "The existence of measures of a given cocycle, II: probability measures." Ergodic Theory and Dynamical Systems 28, no. 5 (October 2008): 1615–33. http://dx.doi.org/10.1017/s0143385707001125.

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AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a probability measure μ on X such that ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere, for every Borel injection ϕ whose graph is contained in E.

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14

Pantsulaia, G. "On an Invariant Borel Measure in Hilbert Space." Bulletin of the Polish Academy of Sciences Mathematics 52, no. 1 (2004): 47–51. http://dx.doi.org/10.4064/ba52-1-5.

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15

Shortt, R. M. "Representation of Borel isomorphism by a probability measure." Proceedings of the American Mathematical Society 104, no. 1 (January 1, 1988): 284. http://dx.doi.org/10.1090/s0002-9939-1988-0958084-2.

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16

Bachman, George, and P. D. Stratigos. "On measure repleteness and support for lattice regular measures." International Journal of Mathematics and Mathematical Sciences 10, no. 4 (1987): 707–24. http://dx.doi.org/10.1155/s0161171287000814.

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Анотація:

The present paper is mainly concerned with establishing conditions which .assure that all lattice regular measures have additional smoothness properties or that simply all two-valued such measures have such properties and are therefore Dirac measures. These conditions are expressed in terms of the general Wallman space. The general results are then applied to specific topological lattices, yielding new conditions for measure compactness, Borel measure compactness, clopen measure repleteness, strong measure compactness, etc. In addition, smoothness properties in the general setting for lattice regular measures are related to the notion of support, and numerous applications are given.

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17

Lee, Keonhee, C. A. Morales, and Bomi Shin. "On the set of expansive measures." Communications in Contemporary Mathematics 20, no. 07 (October 14, 2018): 1750086. http://dx.doi.org/10.1142/s0219199717500869.

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Анотація:

We prove that the set of expansive measures of a homeomorphism of a compact metric space is a [Formula: see text] subset of the space of Borel probability measures equipped with the weak* topology. Next that every expansive measure of a homeomorphism of a compact metric space can be weak* approximated by expansive measures with invariant support. In addition, if the expansive measures of a homeomorphism of a compact metric space are dense in the space of Borel probability measures, then there is an expansive measure whose support is both invariant and close to the whole space with respect to the Hausdorff metric. Henceforth, if the expansive measures are dense in the space of Borel probability measures, the set of heteroclinic points has no interior and the space has no isolated points.

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18

Csiszár, Imre, and F. Matúš. "Convex cores of measures on R d." Studia Scientiarum Mathematicarum Hungarica 38, no. 1-4 (May 1, 2001): 177–90. http://dx.doi.org/10.1556/sscmath.38.2001.1-4.12.

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Анотація:

We define the convex core of a finite Borel measure Q on R d as the intersection of all convex Borel sets C with Q(C) =Q(R d). It consists exactly of means of probability measures dominated by Q. Geometric and measure-theoretic properties of convex cores are studied, including behaviour under certain operations on measures. Convex cores are characterized as those convex sets that have at most countable number of faces.

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19

MILLER, BENJAMIN. "The existence of measures of a given cocycle, I: atomless, ergodic σ-finite measures". Ergodic Theory and Dynamical Systems 28, № 5 (жовтень 2008): 1599–613. http://dx.doi.org/10.1017/s0143385707001113.

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AbstractGiven a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle $\rho : E \rightarrow (0, \infty )$, we characterize the circumstances under which there is a suitably non-trivial σ-finite measure μ on X such that, for every Borel injection ϕ whose graph is contained in E, ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere.

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20

Park, Jaehui. "Toeplitz Operators whose Symbols Are Borel Measures." Journal of Function Spaces 2021 (April 7, 2021): 1–11. http://dx.doi.org/10.1155/2021/5599823.

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Анотація:

In this paper, we are concerned with Toeplitz operators whose symbols are complex Borel measures. When a complex Borel measure μ on the unit circle is given, we give a formal definition of a Toeplitz operator T μ with symbol μ , as an unbounded linear operator on the Hardy space. We then study various properties of T μ . Among them, there is a theorem that the domain of T μ is represented by a trichotomy. Also, it was shown that if the domain of T μ contains at least one polynomial, then T μ is densely defined. In addition, we give evidence for the conjecture that T μ with a singular measure μ reduces to a trivial linear operator.

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21

Szarek, Tomasz. "On typical Markov operators acting on Borel measures." Abstract and Applied Analysis 2005, no. 5 (2005): 489–97. http://dx.doi.org/10.1155/aaa.2005.489.

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It is proved that, in the sense of Baire category, almost every Markov operator acting on Borel measures is asymptotically stable and the Hausdorff dimension of its invariant measure is equal to zero.

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22

Kozarzewski, Piotr A. "On existence of the support of a Borel measure." Demonstratio Mathematica 51, no. 1 (May 1, 2018): 76–84. http://dx.doi.org/10.1515/dema-2018-0010.

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Abstract We present arguments showing that the standard notion of the support of a probabilistic Borel measure is not well defined in every topological space. Our goal is to create a “very inseparable” space and to show the existence of a family of closed sets such that each of them is of full measure, but their intersection is empty. The presented classic construction is credited to Jean Dieudonné and dates back to 1939. We also propose certain, up to our best knowledge, new simplifications.

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23

Shortt, R. M. "Representation of an abstract measure using Borel-isomorphism types." Proceedings of the American Mathematical Society 105, no. 3 (March 1, 1989): 609. http://dx.doi.org/10.1090/s0002-9939-1989-0947317-5.

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24

Zeinal Zadeh Farhadi, Fariba, Mohammad Sadegh Asgari, Mohammad Reza Mardanbeigi, and Mahdi Azhini. "GENERALIZED BESSEL AND FRAME MEASURES." Facta Universitatis, Series: Mathematics and Informatics 35, no. 1 (April 6, 2020): 217. http://dx.doi.org/10.22190/fumi2001217z.

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Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

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25

Grillot, Michele, and Laurent Véron. "Boundary trace of the solutions of the prescribed Gaussian curvature equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 130, no. 3 (June 2000): 527–60. http://dx.doi.org/10.1017/s0308210500000299.

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We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

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26

Kechris, Alexander S. "Amenable versus hyperfinite Borel equivalence relations." Journal of Symbolic Logic 58, no. 3 (September 1993): 894–907. http://dx.doi.org/10.2307/2275102.

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Анотація:

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.

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27

Delladio, S. "Lower semicontinuity and continuity of functions of measures with respect to the strict convergence." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 119, no. 3-4 (1991): 265–78. http://dx.doi.org/10.1017/s0308210500014827.

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SynopsisLet Ω be an open subset of Rn. It is well known that, given a suitable real-valued function f on Ω × Rk and a Rk -valued Borel measure µ on Ω, then one can define a real-valued measurefµ on Ω. The object of this note is to define the Ψ-strict convergence of the Rk-valued Borel measures µj to the Rk-valued Borel measure µ, where Ψ: Ω × Rk → [0, + ∞] is a continuous function which is positively homogeneous and convex in the Rk-variable, and to investigate the lower semicontinuity and continuity of the map µ → fμ with respect to the Ψ-strict convergence; here f is positively homogeneous in the Rk-variable and satisfies one suitable convexity condition (related to Ψ).

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28

Nedic, Mitja. "Characterizations of the lebesgue measure and product measures related to holomorphic functions having non-negative imaginary or real part." International Journal of Mathematics 31, no. 12 (October 10, 2020): 2050102. http://dx.doi.org/10.1142/s0129167x20501025.

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Анотація:

In this paper, we study a class of Borel measures on [Formula: see text] that arises as the class of representing measures of Herglotz-Nevanlinna functions. In particular, we study product measures within this class where products with the Lebesgue measures play a special role. Hence, we give several characterizations of the [Formula: see text]-dimensional Lebesgue measure among all such measures and characterize all product measures that appear in this class of measures. Furthermore, analogous results for the class of positive Borel measures on the unit poly-torus with vanishing mixed Fourier coefficients are also presented, and the relation between the two classes of measures with regard to the obtained results is discussed.

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29

Pantsulaia, Gogi Rauli. "Equipment of Sets with Cardinality of the Continuum by Structures of Polish Groups with Haar Measures." International Journal of Advanced Research in Mathematics 5 (June 2016): 8–22. http://dx.doi.org/10.18052/www.scipress.com/ijarm.5.8.

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It is introduced a certain approach for equipment of sets with cardinality of the continuum by structures of Polish groups with two-sided (left or right) invariant Haar measures. By using this approach we answer positively Maleki’s certain question (2012) what are the real k-dimensional manifolds with at least two different Lie group structures that have the same Haar measure. It is demonstrated that for each diffused Borel probability measure defined in a Polish space (G;ρ;Bρ(G)) without isolated points there exist a metric ρ1and a group operation ⊙ in G such that Bρ(G) = Bρ1(G) and (G;ρ1;Bρ1(G);⊙) stands a compact Polish group with a two-sided (left or right) invariant Haar measure μ , where Bρ(G) and Bρ1(G) denote Borel σ-algebras of subsets of G generated by metrics ρ and ρ1, respectively. Similar approach is used for a construction of locally compact non-compact or non-locally compact Polish groups equipped with two-sided (left or right) invariant quasi-finite Borel measures.

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30

Bartoszynski, Tomek, and Saharon Shelah. "Dual Borel Conjecture and Cohen reals." Journal of Symbolic Logic 75, no. 4 (December 2010): 1293–310. http://dx.doi.org/10.2178/jsl/1286198147.

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31

Hjorth, Greg. "Borel equivalence relations which are highly unfree." Journal of Symbolic Logic 73, no. 4 (December 2008): 1271–77. http://dx.doi.org/10.2178/jsl/1230396917.

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32

Ohta, Haruto, and Ken-ichi Tamano. "Topological spaces whose Baire measure admits a regular Borel extension." Transactions of the American Mathematical Society 317, no. 1 (January 1, 1990): 393–415. http://dx.doi.org/10.1090/s0002-9947-1990-0946425-5.

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33

Milovanovic, Gradimir, Aleksandar Cvetkovic, and Marija Stanic. "A trigonometric orthogonality with respect to a nonnegative Borel measure." Filomat 26, no. 4 (2012): 689–96. http://dx.doi.org/10.2298/fil1204689m.

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Анотація:

In this paper we consider trigonometric polynomials of semi-integer degree orthogonal with respect to a linear functional, defined by a nonnegative Borel measure. By using a suitable vector form we consider the corresponding Fourier sums and reproducing kernels for trigonometric polynomials of semi- integer degree. Also, we consider the Christoffel function, and prove that it satisfies extremal property analogous with the algebraic case.

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34

Wang, Xuejun, Xinghui Wang, Xiaoqin Li, and Shuhe Hu. "Extensions of the Borel–Cantelli lemma in general measure spaces." Journal of Theoretical Probability 27, no. 4 (October 17, 2013): 1229–48. http://dx.doi.org/10.1007/s10959-013-0526-8.

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35

Jorgensen, Palle E. T., and Steen Pedersen. "Spectral theory for borel sets in Rn of finite measure." Journal of Functional Analysis 107, no. 1 (July 1992): 72–104. http://dx.doi.org/10.1016/0022-1236(92)90101-n.

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36

Ihoda, Jaime I. "Strong measure zero sets and rapid filters." Journal of Symbolic Logic 53, no. 2 (June 1988): 393–402. http://dx.doi.org/10.1017/s0022481200028346.

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Анотація:

AbstractWe prove that cons(ZF) implies cons(ZF + Borel conjecture + there exists a Ramsey ultrafilter). We also prove some results on strong measure zero sets from the existence of generalized Luzin sets. We study the relationships between strong measure zero sets and rapid filters on ω.

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37

Živaljević, Boško. "Some results about Borel sets in descriptive set theory of hyperfinite sets." Journal of Symbolic Logic 55, no. 2 (June 1990): 604–14. http://dx.doi.org/10.2307/2274650.

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Анотація:

A remarkable result of Henson and Ross [HR] states that if a function whose graph is Souslin in the product of two hyperfinite sets in an ℵ1 saturated nonstandard universe possesses a certain nice property (capacity) then there exists an internal subfunction of the given one possessing the same property. In particular, they prove that every 1-1 Souslin function preserves any internal counting measure, and show that every two internal sets A and B with ∣A∣/∣B∣ ≈ 1 are Borel bijective. As a supplement to the last-mentioned result of Henson and Ross, Keisler, Kunen, Miller and Leth showed [KKML] that two internal sets A and B are bijective by a countably determined bijection if and only if ∣A∣/∣B∣ is finite and not infinitesimal.In this paper we first show that injective Borel functions map Borel sets into Borel sets, a fact well known in classical descriptive set theory. Then, we extend the result of Henson and Ross concerning the Borel bijectivity of internal sets whose quotient of cardinalities is infinitely closed to 1. We prove that two Borel sets, to which we may assign a counting measure not equal to 0 or ∞, are Borel bijective if and only if they have the same counting measure ≠0, ∞. This, together with the similar characterization for Souslin and measurable countably determined sets, extends the above-mentioned results from [HR] and [KKML].

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38

Mashford, John. "A Spectral Calculus for Lorentz Invariant Measures on Minkowski Space." Symmetry 12, no. 10 (October 15, 2020): 1696. http://dx.doi.org/10.3390/sym12101696.

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This paper presents a spectral calculus for computing the spectra of causal Lorentz invariant Borel complex measures on Minkowski space, thereby enabling one to compute their densities with respect to Lebesque measure. The spectra of certain elementary convolutions involving Feynman propagators of scalar particles are computed. It is proved that the convolution of arbitrary causal Lorentz invariant Borel complex measures exists and the product of such measures exists in a wide class of cases. Techniques for their computation in terms of their spectral representation are presented.

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39

Pawlikowski, Janusz. "Finite support iteration and strong measure zero sets." Journal of Symbolic Logic 55, no. 2 (June 1990): 674–77. http://dx.doi.org/10.2307/2274657.

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Анотація:

AbstractAny finite support iteration of posets with precalibre ℵ1 which has the length of cofinahty greater than ω1 yields a model for the dual Borel conjecture in which the real line is covered by ℵ1 strong measure zero sets.

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40

Kechris, Alexander S. "Countable sections for locally compact group actions." Ergodic Theory and Dynamical Systems 12, no. 2 (June 1992): 283–95. http://dx.doi.org/10.1017/s0143385700006751.

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AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

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41

Magnani, Valentino. "On a measure-theoretic area formula." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 145, no. 4 (July 20, 2015): 885–91. http://dx.doi.org/10.1017/s030821051500013x.

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Анотація:

We review some classical differentiation theorems for measures, showing how they can be turned into an integral representation of a Borel measure with respect to a fixed Carathéodory measure. We focus our attention on the case when this measure is the spherical Hausdorff measure, giving a metric measure area formula. Our aim is to use certain covering derivatives as ‘generalized densities’. Some consequences for the sub-Riemannian Heisenberg group are also pointed out.

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42

MILLER, BENJAMIN D. "On the existence of cocycle-invariant Borel probability measures." Ergodic Theory and Dynamical Systems 40, no. 11 (April 12, 2019): 3150–68. http://dx.doi.org/10.1017/etds.2019.28.

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43

Peralta, Gilbert. "Optimal Borel measure controls for the two-dimensional stationary Boussinesq system." ESAIM: Control, Optimisation and Calculus of Variations 28 (2022): 22. http://dx.doi.org/10.1051/cocv/2022016.

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Анотація:

We analyze an optimal control problem for the stationary two-dimensional Boussinesq system with controls taken in the space of regular Borel measures. Such measure-valued controls are known to produce sparse solutions. First-order and second-order necessary and sufficient optimality conditions are established. Following an optimize-then-discretize strategy, the corresponding finite element approximation will be solved by a semi-smooth Newton method initialized by a continuation strategy. The controls are discretized by finite linear combinations of Dirac measures concentrated at the nodes associated with the degrees of freedom for the mini-finite element.

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44

Yuan, Zhihui. "Multifractal formalism for the inverse of random weak Gibbs measures." Stochastics and Dynamics 20, no. 04 (October 15, 2019): 2050024. http://dx.doi.org/10.1142/s0219493720500240.

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Анотація:

Any Borel probability measure supported on a Cantor set included in [Formula: see text] and of zero Lebesgue measure on the real line possesses a discrete inverse measure. We study the validity of the multifractal formalism for the inverse measures of random weak Gibbs measures. The study requires, in particular, to develop in this context of random dynamics a suitable version of the results known for heterogeneous ubiquity associated with deterministic Gibbs measures.

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45

HOLT, ERIC. "A ratio ergodic theorem for Borel actions of ℤd×ℝk". Ergodic Theory and Dynamical Systems 32, № 2 (16 січня 2012): 675–89. http://dx.doi.org/10.1017/s0143385711001076.

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AbstractWe prove a ratio ergodic theorem for free Borel actions of ℤd×ℝk on a standard Borel σ-finite measure space. The proof employs a lemma by Hochman involving coarse dimension, as well as the Besicovitch covering lemma. Due to possible singularity of the measure, we cannot use functional analytic arguments and therefore use Rudolph’s diffusion of the measure onto the orbits of the action. This diffused measure is denoted μx, and our averages are of the form (1/(μx(Bn)))∫ Bnf∘T−v(x) dμx(v). A Følner condition on the orbits of the action is shown, which is the main tool used in the proof of the ergodic theorem.

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46

Nasr, F. Ben. "On some regularity conditions of Borel measures on R." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 63, no. 2 (October 1997): 218–24. http://dx.doi.org/10.1017/s1446788700000653.

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Анотація:

AbstractThe aim of this paper is to resolve Taylor's question concerning certain regularity conditions on a Borel measure. The proposed solution is given in the framework of Brown, Michon and Peyrière, and Olsen.

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47

Keith, Stephen, and Kai Rajala. "A remark on Poincaré inequalities on metric measure spaces." MATHEMATICA SCANDINAVICA 95, no. 2 (December 1, 2004): 299. http://dx.doi.org/10.7146/math.scand.a-14461.

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Анотація:

We show that, in a complete metric measure space equipped with a doubling Borel regular measure, the Poincaré inequality with upper gradients introduced by Heinonen and Koskela [3] is equivalent to the Poincaré inequality with "approximate Lipschitz constants" used by Semmes in [9].

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48

Pantsulaia, Gogi. "Duality of Measure and Category in Infinite-Dimensional Separable Hilbert Spaceℓ2". International Journal of Mathematics and Mathematical Sciences 30, № 6 (2002): 353–63. http://dx.doi.org/10.1155/s0161171202012371.

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49

Simpson, Stephen G. "Mass Problems and Measure-Theoretic Regularity." Bulletin of Symbolic Logic 15, no. 4 (December 2009): 385–409. http://dx.doi.org/10.2178/bsl/1255526079.

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Анотація:

AbstractA well known fact is that every Lebesgue measurable set is regular, i.e., it includes an Fσ set of the same measure. We analyze this fact from a metamathematical or foundational standpoint. We study a family of Muchnik degrees corresponding to measuretheoretic regularity at all levels of the effective Borel hierarchy. We prove some new results concerning Nies's notion of LR-reducibility. We build some ω-models of RCA0 which are relevant for the reverse mathematics of measure-theoretic regularity.

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50

Ross, David A. "Pushing down infinite Loeb measures." MATHEMATICA SCANDINAVICA 104, no. 1 (March 1, 2009): 108. http://dx.doi.org/10.7146/math.scand.a-15087.

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Анотація:

Sufficient conditions are given under which the standard part map on an arbitrary Hausdorff space can be used to push down an infinite nonstandard measure. This makes it easier to construct standard infinite Borel measures using nonstandard techniques.

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