  This paper introduces a notion of regularity (or irregularity) of the point
at infinity for the unbounded open subset of $\rr^{N}$ concerning second order
uniformly elliptic equations with bounded and measurable coefficients,
according as whether the A-harmonic measure of the point at infinity is zero
(or positive). A necessary and sufficient condition for the existence of a
unique bounded solution to the Dirichlet problem in an arbitrary open set of
$\rr^{N}, N\ge 3$ is established in terms of the Wiener test for the regularity
of the point at infinity. It coincides with the Wiener test for the regularity
of the point at infinity in the case of Laplace equation. From the topological
point of view, the Wiener test at infinity presents thinness criteria of sets
near infinity in fine topology. Precisely, the open set is a deleted
neigborhood of the point at infinity in fine topology if and only if infinity
is irregular.
