The notions of compact convex variation and compact convex subdifferential for the mappings from a segment into a locally convex space (LCS) are studied. In the case of an arbitrary complete LCS, each indefinite Bochner integral has compact variation and each strongly absolutely continuous and compact subdifferentiable a.e. mapping is an indefinite Bochner integral. 0. Introduction and prelimin...

The mean density of a random closed set Θ in R with Hausdorff dimension n is the Radon-Nikodym derivative of the expected measure E[H(Θ ∩ · )] induced by Θ with respect to the usual d-dimensional Lebesgue measure. We consider here inhomogeneous Boolean models with lower dimensional typical grain. Under general regularity assumptions on the typical grain, related to the existence of its Minkowsk...

We consider a discrete-time probabilistic inference system where the conclusions and inputs are probability measures on measurable spaces X,X and Y,Y respectively. We are given a kernel N from X to Y (N : X × Y → [0, 1]), which represents the way information about states of affairs (represented by X) is transmitted to a receptor (represented by Y ). Suppose that at time t = i the conclusion is ...

We give sufficient conditions for a metric space to bilipschitz embed in L1. In particular, if X is a length space and there is a Lipschitz map u : X → R such that for every interval I ⊂ R, the connected components of u−1(I) have diameter ≤ const ·diam(I), then X admits a bilipschitz embedding in L1. As a corollary, the Laakso examples [Laa00] bilipschitz embed in L1, though they do not embed i...

We give conditions on Gromov-Hausdorff convergent inverse systems of metric measure graphs which imply that the measured Gromov-Hausdorff limit (equivalently, the inverse limit) is a PI space i.e., it satisfies a doubling condition and a Poincaré inequality in the sense of Heinonen-Koskela [HK96]. The Poincaré inequality is actually of type (1, 1). We also give a systematic construction of exam...

Let (S,Σ , μ) be a complete positive σ -finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L p(S,Σ , X). Let Σ0 be a sub-σ -algebra of Σ and Y a nonempty locally weakly compact convex subset of X such that span Y and its dual have the Radon–Nikodym property. We prove that L p(S,Σ0, Y ) is...

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilatation structures, the first being a tempered one and...