Entropy and—maybe even more so—mutual information are invaluable tools for analyzing properties of probability distributions, especially in coding theory. While there are general definitions for both concepts available for arbitrary probability distributions, these tend to be hard to work with and the literature (e.g., [CT06]) focuses on either discrete, or continuous random variables. In this ...

Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where “almost everywhere” refers to the Lebesgue measure. Our main result is an extension of this theorem where the Lebesgue measure is replaced by an arbitrary measure μ. In particular we show that the differentiability properties of Lipschitz functions at μ-almost every point ar...

Definition 1. Let N ′ be a Carnot group and N be a subgroup of N ′ with Hausdorff dimension k. A subset E of another Carnot Group (M,dM ) is said to be N -rectifiable if there exists U , a positive measure subset of N , and a Lipschitz map f : U → M such that H k M (E \ f(U)) = 0. We say E is countably N -rectifiable if there exist a countable number of pairs of subsets Ui and Lipschitz maps fi...

A continuous map between topological f : X → Y is said to satisfy the path-lifting property if for any path p : [0, 1] → Y and any x ∈ f(p(0)) there exists a lifting p̃ of the path p with the intitial value x, i.e. there exists a path p̃ such that f ◦ p̃ = p and p̃(0) = x. Similarly, a smooth map between Riemannian manifolds f : X → Y is said to satisfy the rectifiable path-lifting property if the ...