Some results on Sobolev spaces with respect to a measure and applications to a new transport problem

We recall some known and present several new results about Sobolev spaces defined with respect to a measure, in particular a precise pointwise description of the tangent space to this measure in dimension 1. This allows to obtain an interesting, original compactness result which stays open in R^d, d \textgreater{} 1, and can be applied to a new transport problem, with gradient penalization.