标题： 一种混合规范的Carathéodory测度连接勒贝格体积和曲面内容 显示英文标题

标题： A Mixed-Gauge Caratheodory Measure Bridging Lebesgue Volume and Surface Content

摘要： 我们引入了一个单参数族的Borel正则测度，它在$\mathbb{R}^n$上通过引入与尺度无关的边界结构惩罚来增强Lebesgue测度。利用带有混合量规$h_\lambda(r) = r^n + \lambda r^{n-1}$的Carathéodory外测度构造方法对于$\lambda > 0$，得到的测度$\mu_\lambda$在一个单一的$\sigma$可加框架中无缝地结合了$n$维体积与$(n-1)$维表面贡献。 关键结果包括：(i)$\mu_\lambda$是一个度量外测度，所有博雷尔集都是可测的且是博雷尔正则的；(ii) 缩放性质$\mu_\lambda(tE) = t^n \mu_{\lambda/t}(E)$对$t > 0$；(iii) 有界利普希茨区域$\Omega$的定量可比性，其中维数常数$c_n, C_n > 0$满足$c_n (|\Omega| + \lambda \mathcal{H}^{n-1}(\partial \Omega)) \leq \mu_\lambda(\Omega) \leq C_n (|\Omega| + \lambda \mathcal{H}^{n-1}(\partial \Omega))$，直接将$\mu_\lambda$与周长联系起来。这解决了勒贝格测度忽略边界复杂性的问题，同时保持与卡拉西奥多里-豪斯多夫范式的兼容性。潜在应用包括不规则区域上的鲁棒数值积分、图像和形状处理中的周长正则化泛函，以及具有边界意识的概率建模。 示例提供在$\mathbb{R}$和$\mathbb{R}^2$，以及到 Minkowski 内容和有限边界集的链接。 开放问题包括最优常数、BV 空间中的分层公式以及到可测集的扩展。 显示英文摘要

摘要： We introduce a one-parameter family of Borel regular measures on $\mathbb{R}^n$ that enhances Lebesgue measure by incorporating a scale-invariant penalty for codimension-1 boundary structures. Utilizing Carath\'eodory's outer measure construction with the mixed gauge $h_\lambda(r) = r^n + \lambda r^{n-1}$ for $\lambda > 0$, the resulting measure $\mu_\lambda$ seamlessly combines $n$-dimensional volume with $(n-1)$-dimensional surface contributions in a single $\sigma$-additive framework. Key results include: (i) $\mu_\lambda$ is a metric outer measure, with all Borel sets measurable and Borel regular; (ii) the scaling property $\mu_\lambda(tE) = t^n \mu_{\lambda/t}(E)$ for $t > 0$; (iii) quantitative comparability for bounded Lipschitz domains $\Omega$, where dimensional constants $c_n, C_n > 0$ satisfy $c_n (|\Omega| + \lambda \mathcal{H}^{n-1}(\partial \Omega)) \leq \mu_\lambda(\Omega) \leq C_n (|\Omega| + \lambda \mathcal{H}^{n-1}(\partial \Omega))$, directly relating $\mu_\lambda$ to perimeter. This addresses Lebesgue measure's oversight of boundary complexity while preserving compatibility with the Carath\'eodory-Hausdorff paradigm. Potential applications span robust numerical integration on irregular domains, perimeter-regularized functionals in image and shape processing, and boundary-aware probabilistic modeling. Examples are provided in $\mathbb{R}$ and $\mathbb{R}^2$, alongside links to Minkowski content and sets of finite perimeter. Open problems encompass optimal constants, coarea formulas in BV spaces, and extensions to rectifiable sets.