Density and extension of differentiable functions on metric measure spaces

We consider vector valued mappings de ned on metric measure spaces with a measurable differ-entiable structure and study both approximations by nicer mappings and regular extensions of the givenmappings when de ned on closed subsets. Therefore, we propose a rst approach to these problems, largelystudied on Euclidean and Banach spaces during the last century, for rst order differentiable functions de- ned on these metric measure spaces.