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

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 Carathéodory’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 ${\mu }_{\lambda }(tE)=t^n{\mu }_{\lambda \mathrm{/}t}(E)$ for $t>0$; (iii) quantitative comparability for bounded Lipschitz domains $\mathrm{\Omega }$, where dimensional constants $c_n,C_n>0$ satisfy $c_n(\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+\lambda {\mathcal{H}}^{n-1}(\mathrm{\partial }\mathrm{\Omega }))\le {\mu }_{\lambda }(\mathrm{\Omega })\le C_n(\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }+\lambda {\mathcal{H}}^{n-1}(\mathrm{\partial }\mathrm{\Omega }))$, directly relating ${\mu }_{\lambda }$ to perimeter. This addresses Lebesgue measure’s oversight of boundary complexity while preserving compatibility with the Carathéodory-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.