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

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