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 $h_λ(r) = r^n + λr^{n-1}$ for $λ> 0$, the resulting measure $μ_λ$ seamlessly combines $n$-dimensional volume with $(n-1)$-dimensional surface contributions in a single $σ$-additive framework. Key results include: (i) $μ_λ$ is a metric outer measure, with all Borel sets measurable and Borel regular; (ii) the scaling property $μ_λ(tE) = t^n μ_{λ/t}(E)$ for $t > 0$; (iii) quantitative comparability for bounded Lipschitz domains $Ω$, where dimensional constants $c_n, C_n > 0$ satisfy $c_n (|Ω| + λ\mathcal{H}^{n-1}(\partial Ω)) \leq μ_λ(Ω) \leq C_n (|Ω| + λ\mathcal{H}^{n-1}(\partial Ω))$, directly relating $μ_λ$ 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.