We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R, n ≥ 2. To each locally finite Borel measure μ, we associate a function J̃2(μ, x) which uses a weighted sum to record how closely the mass of μ is concentrated near a line in the triples of dyadic cubes containing x. We show that J̃2(μ, ·) < ∞ μ-a.e. is a necessary condition ...

We consider the problem of finding a measure from the given values of its logarithmic potential on the support. It is well known that a solution to this problem is given by the generalized Laplacian. The case of our main interest is when the support is contained in a rectifiable curve, and the measure is absolutely continuous with respect to the arclength on this curve. Then the generalized Lap...

We introduce the notion of density contractor dimension m in a finite dimensional normed space X. If m+1=dimX, this includes area contracting projectors on hyperplanes whose existence was established by H. Busemann. m=2, contractors are an ersatz for such and their existence, here, follows from works D. Burago S. Ivanov. Once available, corresponding Plateau problem admits solution among rectif...

We construct an isometric embedding from Gigli's abstract tangent module into the concrete of a space admitting (weak) Lipschitz differentiable structure, and give two equivalent conditions which characterize when is isomorphism. Together with arguments recent article by Bate--Kangasniemi--Orponen, this equivalence used to show that ${\rm Lip}-{\rm lip}$ -type condition lip} f\le C|Df|$ implies...

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. prove a version Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman Theorem this setting by showing any lower d-regular set \(E \subseteq H\) that $$\begin{aligned} \textrm{diam}(E)^d + \beta ^d(E) \sim \mathscr {H}^d(E) \text {Error}, \end{aligned}$$where \(\beta ^d(E)\) give...

We prove an upper bound on the rank of abelianised revised fundamental group (called “revised first Betti number”) a compact $\mathsf{RCD^{}}(K,N)$ space, in same spirit celebrated Gromov–Gallot number for smooth Riemannian manifold with Ricci curvature bounded below. When synthetic lower is close enough to (negative) zero and aforementioned saturated (i.e. equal integer part $N$, denoted by $\...

The topological views of a measure space provide deep insights. In this paper, the sigma-set algebraic structure is extended in Hausdorff based on locally compactable neighborhood systems without considering strictly (metrized) Borel variety. null extension gives rise to quasi sigma-semiring sigma-neighborhoods, which are rectifiable view Dieudonné n-space. concepts symmetric signed measure, un...

We provide examples of infinitesimally Hilbertian, rectifiable, Ahlfors regular metric measure spaces having pmGH-tangents that are not Hilbertian.