A note on the richness of convex hulls of VC classes.
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Lugosi, Gábor, Mendelson, Shahar, Koltchinskii, Vladimir (2003)
Electronic Communications in Probability [electronic only]
Kharazishvili, A.B. (1996)
Journal of Applied Analysis
Bohdan Aniszczyk (1986)
Colloquium Mathematicae
Piotr Hajłasz (1995)
Annales de l'I.H.P. Analyse non linéaire
Petr Dostál (2002)
Acta Universitatis Carolinae. Mathematica et Physica
Zdena Riečanová (1972)
Časopis pro pěstování matematiky
Duan, Shujuan, Liu, Dan, Tang, Taiman (2009)
Integers
Zeitler, Herbert, Krämer, Alexander (2006)
Mathematica Pannonica
Leif Mejlbro, Flemming Topsoe (1977)
Mathematische Annalen
Albert Raugi (2009)
Annales de l'I.H.P. Probabilités et statistiques
Let be a standard probability space. We say that a sub-σ-algebra of decomposes μ in an ergodic way if any regular conditional probability with respect to andμ satisfies, for μ-almost every x∈X, . In this case the equality , gives us an integral decomposition in “-ergodic” components. For any sub-σ-algebra of , we denote by the smallest sub-σ-algebra of containing and the collection of all setsAin satisfyingμ(A)=0. We say that isμ-complete if . Let be a non-empty family...
Jun Wu (2000)
Acta Arithmetica
1. Introduction. Given x in (0,1], let x = [d₁(x),d₂(x),...] denote the Engel expansion of x, that is, (1) , where is a sequence of positive integers satisfying d₁(x) ≥ 2 and for j ≥ 1. (See [3].) In [3], János Galambos proved that for almost all x ∈ (0,1], (2) He conjectured ([3], P132) that the Hausdorff dimension of the set where (2) fails is one. In this paper, we prove this conjecture: Theorem. . We use L¹ to denote the one-dimensional Lebesgue measure on (0,1] and to denote the Hausdorff...
James Fickett, Jan Mycielski (1979)
Colloquium Mathematicae
Stevo Todorcevic (2004)
Fundamenta Mathematicae
We show that a σ-algebra 𝔹 carries a strictly positive continuous submeasure if and only if 𝔹 is weakly distributive and it satisfies the σ-finite chain condition of Horn and Tarski.
Piotr Niemiec (2012)
Annales Polonici Mathematici
A topological space Y is said to have (AEEP) if the following condition is satisfied: Whenever (X,) is a measurable space and f,g: X → Y are two measurable functions, then the set Δ(f,g) = x ∈ X: f(x) = g(x) is a member of . It is shown that a metrizable space Y has (AEEP) iff the cardinality of Y is not greater than .
H. W. Pu, J. P. Spencer (1971)
Colloquium Mathematicae
Anzelm Iwanik (1989)
Acta Universitatis Carolinae. Mathematica et Physica
Giuseppa Riccobono (1997)
Mathematica Bohemica
In this paper, we define a -integral, i.e. an integral defined by means of partitions of unity, on a suitable compact metric measure space, whose measure is compatible with its topology in the sense that every open set is -measurable. We prove that the -integral is equivalent to -integral. Moreover, we give an example of a noneuclidean compact metric space such that the above results are true.
Stephen Scheinberg (2019)
Commentationes Mathematicae Universitatis Carolinae
This note contains a proof of the existence of a one-to-one function of onto itself with the following properties: is a rational-linear automorphism of , and the graph of is a non-measurable subset of the plane.
Luca Esposito, Nicola Fusco, Cristina Trombetti (2005)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if is a set with small anisotropic isoperimetric deficit, then is “close” to the Wulff shape set.
S.G. Wayment, L. Hatta (1973)
Journal für die reine und angewandte Mathematik