For all $1⩽m⩽n-1$, we investigate the interaction of locally finite measures in ${\mathbb{R}}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $\mu$, which are carried by Lipschitz graphs in the sense that there exist graphs ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,\cdots$ such that $\mu ({\mathbb{R}}^n\setminus {\bigcup }_1^{\infty }{\mathrm{\Gamma }}_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, for example, for the restrictions of $m$-dimensional Hausdorff measure ${\mathcal{H}}^m$ to $E\subseteq {\mathbb{R}}^n$ with $0<{\mathcal{H}}^m(E)<\infty$. However, an example of Csörnyei, Käenmäki, Rajala, and Suomala shows that classical methods are insufficient to detect when a general measure charges a Lipschitz graph. To develop a characterization of Lipschitz graph rectifiability for arbitrary Radon measures, we look at the behavior of coarse doubling ratios of the measure on dyadic cubes that intersect conical annuli. This extends a characterization of graph rectifiability for pointwise doubling measures by Naples by mimicking the approach used in the characterization of Radon measures carried by rectifiable curves by Badger and Schul.

中文翻译：

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氡测量和 Lipschitz 图

对所有人 $1⩽米⩽n-1$，我们研究了局部有限测度的相互作用 ${\mathbb{电阻}}^n$ 与家人 $米$维 Lipschitz 图。例如，我们描述氡测量$\mu$，在存在图的意义上，它们由 Lipschitz 图携带 ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2,\cdots$ 以至于 $\mu ({\mathbb{电阻}}^n\setminus {\bigcup }_1^{\infty }{\mathrm{\Gamma }}_{\mathrm{一世}})=0$，仅使用可数的多个评估。几何测度理论中的这个问题在较小的测度类别中进行了经典的研究，例如，对于$米$维豪斯多夫测度 ${\mathcal{H}}^{米}$ 至 $乙\subseteq {\mathbb{电阻}}^n$ 和 $0<{\mathcal{H}}^{米}(乙)<\infty$. 然而，Csörnyei、Käenmäki、Rajala 和 Suomala 的一个例子表明，经典方法不足以检测一般度量何时对 Lipschitz 图充电。为了开发任意氡测量的 Lipschitz 图可修正性的特征，我们研究了与圆锥环相交的二元立方体上测量的粗倍率的行为。这通过模仿由 Badger 和 Schul 对可整流曲线所承载的氡量度进行表征所使用的方法，扩展了 Naples 对逐点加倍措施的图可整流性的表征。