Multiscale Analysis of 1-rectifiable Measures II: Characterizations
نویسندگان
چکیده
منابع مشابه
Multiscale Analysis of 1-rectifiable Measures Ii: Characterizations
A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean space for all n ≥ 2 in terms of positivity of the lower density and finiteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which μ admits approximate tangent l...
متن کاملMultiscale Analysis of 1-rectifiable Measures: Necessary Conditions
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 ...
متن کاملTwo Sufficient Conditions for Rectifiable Measures
We identify two sufficient conditions for locally finite Borel measures on R to give full mass to a countable family of Lipschitz images of R. The first condition, extending a prior result of Pajot, is a sufficient test in terms of L affine approximability for a locally finite Borel measure μ on R satisfying the global regularity hypothesis lim sup r↓0 μ(B(x, r))/r <∞ at μ-a.e. x ∈ R to be m-re...
متن کاملAxiomatic Characterizations of Information Measures
Axiomatic characterizations of Shannon entropy, Kullback I-divergence, and some generalized information measures are surveyed. Three directions are treated: (A) Characterization of functions of probability distributions suitable as information measures. (B) Characterization of set functions on the subsets of {1, . . . , N} representable by joint entropies of components of an N -dimensional rand...
متن کاملRIESZ s - EQUILIBRIUM MEASURES ON d - RECTIFIABLE SETS
Let A be a compact set in Rp of Hausdorff dimension d. For s ∈ (0, d), the Riesz s-equilibrium measure μs is the unique Borel probability measure with support in A that minimizes Is(μ) := " 1 |x − y|s dμ(y)dμ(x) over all such probability measures. If A is strongly (Hd , d)-rectifiable, then μs converges in the weak-star topology to normalized d-dimensional Hausdorff measure restricted to A as s...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Analysis and Geometry in Metric Spaces
سال: 2017
ISSN: 2299-3274
DOI: 10.1515/agms-2017-0001