Fractal and multifractal dimensions of prevalent measures Lars Olsen 28A80multifractalsHausdorff dimensionpacking dimensionlocal dimensionprevalenceshyness Let $K$ be a compact subset of $\mathbb{R}^{d}$ and write $\mathcal{P}(K)$ for the family of Borel probability measures on $K$. In this paper we study different fractal and multifractal dimensions of measures $\mu$ in $\mathcal{P}(K)$ that are generic in the sense of prevalence. We first prove a general result, namely, for an arbitrary ``dimension'' function $\Delta: \mathcal{P}(K) \to \mathbb{R}$ satisfying various natural scaling and monotonicity conditions, we obtain a formula for the ``dimension'' $\Delta(\mu)$ of a prevalent measure $\mu$ in $\mathcal{P}(K)$; this is the content of Theorem 1.1. By applying Theorem 1.1 to appropriate choices of $\Delta$ we obtain the following results: \begin{itemize}\item By letting $\Delta(\mu)$ equal the (lower or upper) local dimension of $\mu$ at a point $x \in K$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the (lower and upper) local dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item By letting $\Delta(\mu)$ equal the multifractal spectrum of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the multifractal spectrum of a prevalent measure $\mu$ in $\mathcal{P}(K)$. \item Finally, by letting $\Delta(\mu)$ equal the Hausdorff or packing dimension of $\mu$ and applying Theorem 1.1 to this particular choice of $\Delta$, we compute the Hausdorff and packing dimension of a prevalent measure $\mu$ in $\mathcal{P}(K)$.\end{itemize} 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.3991 10.1512/iumj.2010.59.3991 en Indiana Univ. Math. J. 59 (2010) 661 - 690 state-of-the-art mathematics http://iumj.org/access/