2025/08/25 by Thakur, Yash
#FOS: Mathematics #General Mathematics (math.GM)
paper · doi:10.48550/arxiv.2508.18011
We introduce a one-parameter family of Borel regular measures on ℝn 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 hλ(r) = rn + λrn-1 for λ> 0, the resulting measure μλ seamlessly combines n-dimensional volume with (n-1)-dimensional surface contributions in a single σ-additive framework. Key results include: (i) μλ is a metric outer measure, with all Borel sets measurable and Borel regular; (ii) the scaling property μλ(tE) = tn μλ/t(E) for t > 0; (iii) quantitative comparability for bounded Lipschitz domains Ω, where dimensional constants cn, Cn > 0 satisfy cn (|Ω| + λHn-1(∂ Ω)) ≤ μλ(Ω) ≤ Cn (|Ω| + λHn-1(∂ Ω)), directly relating μλ 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 ℝ and ℝ2, 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.