In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a non-stationary process. To overcome this problem we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study L2-convergence, non-degeneracy and continuity of the limit process. Establishing a power law for its moments we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.