  In this paper, we study the global regularity estimates in Lorentz spaces for
gradients of solutions to quasilinear elliptic equations with measure data of
the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x,
\nabla u))&=& \mu \quad \text{in} ~\Omega, u&=&0 \quad \text{on}~ \partial
\Omega, \end{array}\right. \end{eqnarray*} where $\mu$ is a finite signed Radon
measure in $\Omega$, $\Omega \subset \mathbb{R}^n$ is a bounded domain such
that its complement $\mathbb{R}^n\backslash\Omega$ is uniformly $p$-thick and
$\mathcal{A}$ is a Carath\'eodory vector valued function satisfying growth and
monotonicity conditions for the strongly singular case $1<p\leq
\frac{3n-2}{2n-1}$. Our result extends the earlier results \cite{55Ph0,Tran19}
to the strongly singular case $1<p\leq \frac{3n-2}{2n-1}$ and a recent result
\cite{HP} by considering rough conditions on the domain $\Omega$ and the
nonlinearity $\mathcal{A}$.
