Abstract 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 L2 gauge the extent to which μ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between μ and 1-dimensional Hausdorff measure H1. We also characterize purely 1-unrectifiable Radon measures, i.e. locally finite measures that give measure zero to every finite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L2 variant of P. Jones’ traveling salesman construction, which is of independent interest.

中文翻译：

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1-可纠正措施的多尺度分析 II：表征

摘要 如果存在补集为零的有限长度曲线的可数并集，则测度是 1-可修正的。我们根据几何平方函数的较低密度和有限性的正性来表征 n 维欧几里得空间中所有 n ≥ 2 的 1-可整流氡测量 μ，它松散地说，在 L2 规范中记录 μ 承认的程度近似切线，或具有快速增长的密度比，沿着它的支持。与 Besicovitch、Morse 和 Randolph 以及 Moore 的经典定理相比，我们不假设 μ 和一维 Hausdorff 测度 H1 之间存在先验关系。我们还描述了纯粹的 1-不可校正氡测量，即局部有限测量，为每条有限长度曲线给出测量零。这种形式的特征最初是由 P. Jones 推测存在的。在此过程中，我们开发了 P. Jones 旅行商结构的 L2 变体，这是独立的兴趣。