Let K be a compact subset of R~d and write P(K) for the family of Borel probability measures on K. In this paper we study different fractal and multifractal dimensions of measures μ in P(K) that are generic in the sense of prevalence. We first prove a general result, namely, for an arbitrary "dimension" function Δ : P(K)→R satisfying various natural scaling and monotonicity conditions, we obtain a formula for the "dimension" Δ (μ) of a prevalent measure μ in P(K); this is the content of Theorem 1.1. By applying Theorem 1.1 to appropriate choices of Δ we obtain the following results.
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