In this paper we prove that, given s ≥ 0, and a Borel non zero measure μ in Rm, if for μ-almost every x ∈ Rm the limit mathematical equation presented exists and 0 < lim sup r→0μ(B(x; r)) /rs < ∞, then s in an integer. In particular, if E ⊂ Rm is a set with positive and bounded s-dimensional Hausdorff measure Hs and for H s-almost every x ∈ E the limit mathematical equation presented exists, then s is an integer. ©, Vol. 59, No. 1 (2010).
- Hausdorff measures
- Riesz transforms