Integral Metric Spaces and Scalar Curvature Recovery

We introduce the notion of an integral metric space, a triplet (X, d, μ) where μ is a non-atomic probability measure and the classical triangle inequality is replaced by an integral analogue: the distance d(x,y) is bounded above by F(x) + F(y), where F(x) = ∫_X d(x,z) dμ(z) is the central functional measuring the average distance from x to the space. This framework strictly generalizes classical metric spaces. We establish the axioms, verify their internal coherence through eight structural properties, prove that on any space with at least three points an admissible measure always exists, and show stability under products. We then develop the local extension, in which each point carries its own measure μ_x — the analogue of the Riemannian metric tensor g_ij(x). We prove that on any smooth Riemannian manifold (M,g) of dimension n, the local central functional satisfies F^ε(x) = n/(n+1) · ε − (n+3)/(6(n+1)(n+2)) · Scal(x) · ε³ + O(ε⁴), recovering the scalar curvature from average distances alone. This shows that Riemannian geometry is a special case of the local integral framework.