In Version 9 currently, we can do (using the Undocumented form of Integrate):
Integrate[Boole[z >= 0] z, {x, y, z} ∈ Sphere[{0, 0, 0}, 4]]
64 Pi
Note: This is undocumented behaviour and functionality may change or behave differently in newer versions of Mathematica so use with caution e.g. as noted by Szabolcs in the comment, Sphere in V10 denotes a surface, whereas in V9 it represents a volume. So we should keep this in mind or we can get as fancy as V10 and use BallRegion (equivalent to Ball as Szabolcs did). To do this, first load the Region context
Graphics`Region`RegionInit[]
Then:
Integrate[ Boole[z >= 0] z, {x, y, z} ∈ BallRegion[{0, 0, 0}, 4]]
64 Pi