The Arabs arrived first; after some considerable period of time, mercenaries were hired by the Byzantines to drive them out. After that, some Normans arrived and carved out a short-lived kingdom in Sicily and if memory serves a small chunk of the Italian boot. Wanting the best for his kids, King Roger (”fweee Woger!”) hired Arab tutors and the whole thing went all kumbaya after that.
The (purported?) tomb of Hrolf the Ganger has a very-postmortem inscription referencing the various dynasties which had sprung from him, and the Sicily kingdom was one of them.
Actually the Normans conquered a good portion of southern Italy first, concluding with the conquest of Bari in 1071—the last Byzantine possession in the West. The next year the Norman conquest of Sicily began with the capture of Palermo.