A Fubini-type theorem for Hausdorff dimension

It is well known that a classical Fubini theorem for Hausdorff dimension cannot hold; that is, the dimension of the intersections of a fixed set with a parallel family of planes do not determine the dimension of the set. Here we prove that a Fubini theorem for Hausdorff dimension does hold modulo sets that are small on all Lipschitz graphs. We say that G ⊂ℝ^k×ℝ^n is Γ k -null if for every Lipschitz function f:ℝ^k→ℝ^n the set {t ∈ℝ^k:(t,f(t)) ∈ G} has measure zero. We show that for every Borel set E ⊂ℝ^k×ℝ^n with dim (proj_ℝ^kE) = k there is a Γ k -null subset G ⊂ E such that (E\ G) = k + ess - sup (E_t) where ess- sup(dim E t ) is the essential supremum of the Hausdorff dimension of the vertical sections {E_t} _t ∈ℝ^k of E . In addition, we show that, provided that E is not Γ k -null, there is a Γ k -null subset G ⊂ E such that for F = E G , the Fubini property holds, that is, dim ( F ) = k + ess-sup(dim F t ). We also obtain more general results by replacing ℝ k by an Ahlfors–David regular set. Applications of our results include Fubini-type results for unions of affine subspaces, connection to the Kakeya conjecture and projection theorems.