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 $\mu$, in particular a precise pointwise description of the tangent space to $\mu$ in dimension 1. This allows to obtain an interesting, original compactness result which stays open in $\mathbb R^d$, $d>1$, and can be applied to a new transport problem, with gradient penalization.