  We study the existence of very weak solutions to a system
\[\begin{cases}-\mathrm{div}
\mathcal{A}(x,D\mathbf{u})=\mathbf{\mu}\quad\text{in }\ \Omega,
  \mathbf{u}=0\quad\text{on }\ \partial\Omega\end{cases} \]
  with a datum $\mathbf{\mu}$ being a vector-valued bounded Radon measure and
$\mathcal{A}$ having measurable dependence on the spacial variable and Orlicz
growth with respect to the second variable. We are {\em not} restricted to the
superquadratic case.
  For the solutions and their gradients we provide regularity estimates in the
generalized Marcinkiewicz scale. In addition, we show a precise sufficient
condition for the solution to be a~Sobolev function.
