We investigate the functional quantization problem for stochastic processes with respect to L~p(IR~d,μ)-norms, where μ is a fractal measure namely, μ is self-similar or a homogeneous Cantor measure. The derived functional quantization upper rate bounds are universal depending only on the mean-regularity index of the process and the quantization dimension of μ and as universal rates they are optimal. Furthermore, for arbitrary Borel probability measures μ we establish a (nonconstructive) link between the quantization errors of and the functional quantization errors of the process in the space L~p(IR~d, μ).
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