We consider the ensemble of curves {γα, N: α∈(0, 1], N∈ℕ} obtained by linearly interpolating the values of the normalized theta sum N−1/2∑n=0N'−1exp(πin2α), 0≤N'<N. We prove the existence of limiting finite-dimensional distributions for such curves as N→∞, when α is distributed according to any probability measure λ, absolutely continuous w.r.t. the Lebesgue measure on [0, 1]. Our Main Theorem generalizes a result by Marklof [Duke Math. J.97 (1999) 127–153] and Jurkat and van Horne [Duke...

In this paper the problem of the existence of an inverse (or projective) limit measure ${\mu }^{\text{'}}$ of an inverse system of measure spaces $(X_i,{\mu }_i)$ is approached by obtaining first a measure $\tilde{\mu }$ on the whole product space ${\prod }_{i\in I}{X}_i$.The measure $\tilde{\mu }$ will have many of the properties of a limit measure provided only that the measures ${\mu }_i$ possess mild regularity properties.It is shown that ${\mu }^{\text{'}}$ can only exist when $\tilde{\mu }$ is itself a “limit” measure in a more general sense, and that ${\mu }^{\text{'}}$ must then be the restriction of $\tilde{\mu }$ to the projective limit...

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 $ca(ℬ,\lambda ,X)\setminus M_{\sigma }$, 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 ${\omega }_1\times {\omega }_2$ and vn on ${\omega }_2$ satisfying the constraint $v_n\left(y\right)=\frac{1}{|{\omega }_1|}{\int }_{{\omega }_1}{u}_n(x,y)dx$. 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 $\mu :\mathbf{H}\left(T\right)\to E$, à un espace de fonctions scalaires ${\mathbf{L}}^1\left(\mu \right)$, et l’étude détaillée de ce prolongement. L’outil essentiel est la “semi-variation” associée à $\mu$ dans le cas où $E$ 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 ${E_i}_i$ and ${F_j}_j$ are dust-like graph-directed sets satisfying the transitivity condition, then $E_{i₁}$ and $E_{i₂}$ are Lipschitz equivalent, and $E_i$ and $F_j$ are quasi-Lipschitz equivalent when they have the same Hausdorff dimension.

We give a sufficient condition for a curve $\gamma :ℝ\to {ℝ}^n$ to ensure that the $1$-dimensional Hausdorff measure restricted to $\gamma$ is locally monotone.