  We prove that the Dirichlet problem for degenerate elliptic equations
$\mathrm{div}(A \nabla u) = 0$ in the upper half-space $(x,t)\in
\mathbb{R}^{n+1}_+$ is solvable when $n\geq2$ and the boundary data is in
$L^p_\mu(\mathbb{R}^n)$ for some $p<\infty$. The coefficient matrix $A$ is only
assumed to be measurable, real-valued and $t$-independent with a degenerate
bound and ellipticity controlled by an $A_2$-weight $\mu$. It is not required
to be symmetric. The result is achieved by proving a Carleson measure estimate
for all bounded solutions in order to deduce that the degenerate elliptic
measure is in $A_\infty$ with respect to the $\mu$-weighted Lebesgue measure on
$\mathbb{R}^n$. The Carleson measure estimate allows us to avoid applying the
method of $\epsilon$-approximability, which simplifies the proof obtained
recently in the case of uniformly elliptic coefficients. The results have
natural extensions to Lipschitz domains.
