Quasiadditivity and measure property of capacity and the tangential boundary behavior of harmonic functions

Abstract

We show that if a set E E is dispersely decomposed into subsets, then the capacity of E E is comparable to the summation of the capacities of the subsets. From this fact it is derived that the Lebesgue measure of a certain expanded set is estimated by the capacity of E E . These properties hold for classical capacities, L p L^{p} -capacities and energy capacities of general kernels. The estimation is applied to the boundary behavior of harmonic functions. We introduce a boundary thin set and show a fine limit type boundary behavior of harmonic functions. We show that a thin set does not meet essentially Nagel-Stein and Nagel-Rudin-Shapiro type approaching regions at almost all bounary points.