A new approach to Sobolev spaces in metric measure spaces

Let (X,dX,μ)(X,dX,μ) be a metric measure space where XX is locally compact and separable and μμ is a Borel regular measure such that 0<μ(B(x,r))<∞0<μ(B(x,r))<∞ for every ball B(x,r)B(x,r) with center x∈Xx∈X and radius r>0r>0. We define XX to be the set of all positive, finite non-zero regular Borel measures with compact support in XX which are dominated by μμ, and M=X∪{0}M=X∪{0}. By introducing a kind of mass transport metric dMdM on this set we provide a new approach to first order Sobolev spaces on metric measure spaces, first by introducing such for functions F:X→RF:X→R, and then for functions f:X→[−∞,∞]f:X→[−∞,∞] by identifying them with the unique element Ff:X→RFf:X→R defined by the mean-value integral:Ff(η)=1‖η‖∫fdη. In the final section we prove that the approach gives us the classical Sobolev spaces when we are working in open subsets of Euclidean space RnRn with Lebesgue measure.