Start with a compact set K ⊂ Rd. This
has a random number of daughter sets, each of which is a (rotated and scaled)
copy of K and all of which are inside K. The random mechanism for
producing daughter sets is used independently on each of the daughter sets to
produce the second generation of sets, and so on, repeatedly. The random
fractal set F is the limit, as n goes to ∞, of the union
of the nth generation sets. In addition, K has a (suitable,
random) mass which is divided randomly between the daughter sets, and this
random division of mass is also repeated independently, indefinitely. This
division of mass will correspond to a random self-similar measure on F.
The multifractal spectrum of this measure is studied here. Our main
contributions are dealing with the geometry of realisations in
Rd and drawing systematically on known results for
general branching processes. In this way we generalise considerably the results
of Arbeiter and Patzschke (1996) and Patzschke (1997).