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It is proved that if X is infinite-dimensional, then there exists an infinite-dimensional space of X-valued measures which have infinite variation on sets of positive Lebesgue measure. In term of spaceability, it is also shown that , the measures with non-σ-finite variation, contains a closed subspace. Other considerations concern the space of vector measures whose range is neither closed nor convex. All of those results extend in some sense theorems of Muñoz Fernández et al. [Linear Algebra Appl....
Let be a mapping from a metric space X to a metric space Y, and let α be a positive real number. Write dim (E) and Hs(E) for the Hausdorff dimension and the s-dimensional Hausdorff measure of a set E. We give sufficient conditions that the equality dim (f(E)) = αdim (E) holds for each E ⊆ X. The problem is studied also for the Cantor ternary function G. It is shown that there is a subset M of the Cantor ternary set such that Hs(M) = 1, with s = log2/log3 and dim(G(E)) = (log3/log2) dim (E), for...
Representation of bounded and compact linear operators in the Banach space of regulated functions is given in terms of Perron-Stieltjes integral.
We give necessary and sufficient conditions
which characterize the Young measures associated to
two oscillating sequences of functions, un on and
vn on satisfying the constraint . Our study is motivated by
nonlinear effects induced by homogenization. Techniques based on
equimeasurability and rearrangements are employed.
Cet article concerne une méthode nouvelle de prolongement d’une mesure de Radon , à un espace de fonctions scalaires , et l’étude détaillée de ce prolongement. L’outil essentiel est la “semi-variation” associée à dans le cas où est un espace normé, une notion qui a son origine à la fois dans la semi-variation ensembliste de Bartle, Dunford et Schwartz (Canad. J. of Math., t. 7 (1955), 289-305), (New York, London, Interscience Publishers, 1958), et dans l’intégrale supérieure essentielle de...
This paper studies the geometric structure of graph-directed sets from the point of view of Lipschitz equivalence. It is proved that if and are dust-like graph-directed sets satisfying the transitivity condition, then and are Lipschitz equivalent, and and are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.
We give a sufficient condition for a curve to ensure that the -dimensional Hausdorff measure restricted to is locally monotone.
We prove that the 1-dimensional Hausdorff measure restricted to a simple real analytic curve , , is locally 1-monotone.
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