Menger Curvature and Rectifiability 833

where H1 is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger curvature of the triple (x, y, z). A Borel set E ⊂ Rn is said to be “purely unrectifiable” if for any Lipschitz function γ : R → Rn, H1(E ∩ γ(R)) = 0 whereas it is said to be rectifiable if there exists a countable family of Lipschitz functions γi : R → Rn such that H1(E\ ∪i γi(R)) = 0. It may be seen from this definition that any 1-set E (that is, E Borel and 0 < H1(E) <∞) can be decomposed into two subsets