Measure data systems with Orlicz growth

We study the existence of very weak solutions to a system $$\{\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\mathcal{A}(x,D\mathbf{u})=\mathbf{\mu }\text{in }\mathrm{\Omega },\mathbf{u}=0\text{on }\mathrm{\partial }\mathrm{\Omega }\end{array}$$ 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.