Nearly flat almost monotone measures are big pieces of Lipschitz graphs, J. Geom. Anal., 12(1), 2002. 29-61.
A concentrated (ξ,m) almost monotone measure in Rn is a Radon measure φ satisfying the two following conditions: (1) Θm(φ,x) ≥ 1 for every x ∈ spt(φ) and (2) for every x ∈ Rn the ratio exp[ξ(r)]r-mφ(B(x,r)) is nondecreasing as a function of r > 0. Here ξ is a nondecreasing function such that ξ(0+)=0. We prove that there is a relatively open dense set Reg(φ) ⊂ spt(φ) such that at each x ∈ Reg(φ) the support of φ has the following regularity property: given ε > 0 and λ > 0, there is an m dimensional subspace W of Rn and a λ Lipschitz function from x + W to the orthogonal of W so that 100-ε % of spt(φ) ∩ B(x,r) coincides with the graph of f, at some scale r > 0, depending on x, ε and λ.