The authors discuss the concept of robust solution to the measure driven differential inclusion \[ dx(t)\in F(t,x(t))dt+ G(t,x(t))\mu(dt)\quad\text{on }[0,1],\;x(0)=b. \] Here, \(F\) and \(G\) are given multivalued mappings, the initial value \(b\) is a given point in \(\mathbb{R}^n\), and \(\mu\) is some nonnegative scalar-valued measure on the Borel subsets of \([0,1]\). The case in which \(F\) and \(G\) are single-valued mappings, has been previously considered by \textit{G. Dal Maso} and \textit{F. Rampazzo} [Differ. Integral Equ. 4, No. 4, 739-765 (1991; Zbl 0731.34087)]. One of the issues raised by the present paper concerns the stability properties of robust solutions: under suitable assumptions, it is shown that robust solutions to perturbations of a nominal measure driven differential inclusion yield a solution to the nominal differential inclusion in the limit.