We consider reaction-diffusion systems with merely measurable reaction terms to cover the possibility of discontinuities. Solutions of such problems are defined as solutions to appropriate differential inclusions which, in an abstract form, lead to evolution inclusions of the form $u' \in - A u + F(t,u)$ on $[0,T], u(0)=u_{0},$ where $A$ is $m$-accretive and $F$ is of upper semicontinuous type. While such problems, in general, can exhibit non-existence of solutions, the present paper shows that especially for $m$-completely accretive $A$, and under reasonable assumptions on $F$, mild solutions do exist.