Structure of measures in Lipschitz differentiability spaces

We prove the equivalence of two seemingly very di erent ways of generalising Rademacher's theorem to metric measure spaces. The rst was introduced by Cheeger and is based upon di erentiation with respect to another, xed, chart func- tion. The second approach is new for this generality and originates in some ideas of Alberti. It is based upon forming partial derivatives along a very rich structure of Lipschitz curves, analogous to the di erentiability theory of Euclidean spaces. By examining this structure further, we naturally arrive to several descriptions of Lipschitz di erentiability spaces.