Fractal and Multifractal Dimensions of Prevalent Measures

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 mu in T(K) that are generic in the sense of prevalence. We first prove a general result, namely, for an arbitrary "dimension" function Delta : P(K) -> R. satisfying various natural scaling and monotonicity conditions, we obtain a formula for the "dimension" Delta(mu) of a prevalent measure mu in P(K); this is the content of Theorem 1.1. By applying Theorem 1.1 to appropriate choices of Delta we obtain the following results: By letting Delta(mu) equal the (lower or upper) local dimension of mu at a point x is an element of K and applying Theorem 1.1 to this particular choice of mu, we compute the (lower and upper) local dimension of a prevalent measure mu in Delta(K). By letting Delta(mu) equal the multifractal spectrum of mu and applying Theorem 1.1 to this particular choice of mu, we compute the multifractal spectrum of a prevalent measure mu in T(K). Finally, by letting Delta(mu) equal the Hausdorff or packing dimension of mu and applying Theorem 1.1 to this particular choice of mu, we compute the Hausdorff and packing dimension of a prevalent measure mu in T(K). Perhaps surprisingly, in all cases our results are very different from the corresponding results for measures that are generic in the sense of Baire category.