A NEW PROOF OF FEDERER’S STRUCTURE THEOREM FOR k-DIMENSIONAL SUBSETS OF R

If X is a subset of R , let Hk(X) denote the k-dimensional hausdorff measure of X . We write Ik(X) = 0 if H(πKX) = 0 for almost every k-plane K in R , where πK : R → K is orthogonal projection. Otherwise we write Ik(X) > 0. This paper gives a proof of the following theorem. 1.1. Structure Theorem. Let X be a set in R with Hk(X) < ∞ and Ik(X) > 0. Then there is a k-dimensional C submanifold M with Hk(M ∩X) > 0. Let C be the class of countable unions of k-dimensional C submanifolds of R . Since C is closed under countable unions, there is an S ∈ C that maximizes Hk(S ∩X). Letting Y = S ∩X and applying the structure theorem to Z = X \ Y , we get 1.2. Corollary. Suppose X ⊂ R and Hk(X) < ∞. Then X = Y ∪Z, where Y is the portion of X contained in a countable union of k-dimensional C submanifolds and where H(πKZ) = 0 for almost every k-plane K ⊂ R .