We generalize Bourgain's theorem on the two dimensional Kakeya maximal function by proving norm estimates with respect to measures satisfying certain conditions. We use this to extend the classical result of Davies on the Hausdorff dimension of Kakeya sets in the plane.
"Norm Estimates for the Kakeya Maximal Function with Respect to General Measures." Real Anal. Exchange 27 (2) 563 - 572, 2001/2002.