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

Abstract

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.