Abstract. We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.

We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable structure the measure need no longer be pointwise doubling.

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Abstract. We construct a purely unrectifiable set of finite H 1 -measure in every infinite dimensional separable Banach space X whose image under every 0 = x * ∈ X * has positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.