We show that a Radon measure μ in R which is absolutely continuous with respect to the n-dimensional Hausdorff measure H is n-rectifiable if the so called Jones’ square function is finite μ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely contin...

For d ≥ 2, we construct a doubling measure ν on Rd and a rectifiable curve Γ such that ν(Γ) > 0.

Several criteria for the quasianaliticity of Carleman classes at a boundary point of a Jordan domain with rectifiable boundary are found.

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...

Geometric measure theory studies properties of measures, functions and sets. In this note, we provide a basic introduction of geometric measure theory. We will study elementary properties of Hausdorff measures, Lipschitz functions and countably rectifiable sets. To make this note easily accessible, we confine our discussions in Euclidean spaces. Important topics include the area and coarea form...

We characterise rectifiable subsets of a complete metric space $X$ in terms local approximation, with respect to the Gromov--Hausdorff distance, by an $n$-dimensional Banach space. In fact, if $E\subset X$ $\mathcal{H}^n(E)<\infty$ and has positive lower density almost everywhere, we prove that it is sufficient that, at every point each sufficiently small scale, $E$ approximated bi-Lipschitz im...

For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K) > 0, we prove that whenever s > α, any sequence of weighted minimal Riesz s-energy configurations ωN = {x i,N} N i=1 on K (for ‘nice’ weights) is quasi-uniform in the sense that the ratios of its mesh norm to separation distance remain bounded as N grows large. Furthermore, if K is an α-rectifiable com...

In this paper we prove theorems for convergence of discrete solutions of the Plateau problem under the assumption that the contour is rectifiable.