  One of the basic aims of this paper is to study the relationship between the
geometry of ``hypersurface like'' subsets of Euclidean space and the properties
of the measures they support. In this context we show that certain doubling
properties of a measure determine the geometry of its support. A Radon measure
is said to be doubling with constant C if C times the measure of the ball of
radius r centered on the support is greater than the measure of the ball of
radius 2r and the same center. We prove that if the doubling constant of a
measure on \R^{n+1} is close to the doubling constant of the n-dimensional
Lebesgue measure then its support is well approximated by n-dimensional affine
spaces, provided that the support is relatively flat to start with. Primarily
we consider sets which are boundaries of domains in \R^{n+1}. The n-dimensional
Hausdorff measure may not be defined on the boundary of a domain in R^{n+1}.
Thus we turn our attention to the harmonic measure which is well behaved under
minor assumptions. We obtain a new characterization of locally flat domains in
terms of the doubling properties of their harmonic measure. Along these lines
we investigate how the ``weak'' regularity of the Poisson kernel of a domain
determines the geometry of its boundary.
