Let f be a diffeomorphism of a manifold preserving a Borel probability measure $\mu$ having no zero Lyapunov exponents and $\mathcal{W}^s$ a set with asymptotically attracting properties. An upper bound on the Hausdorff dimension of $\mathcal{W}^s$ is given in terms of the Lyapunov exponents and the measure-theoretic entropy. For a surface diffeomorphism, in particular, this upper bound relates, roughly speaking, to the dimension of $\mu$ in the direction of the unstable subspace. Furthermore, we apply this result to establish the equivalence of several properties of $\mu$: its statistical properties, its physical observability and its dimension-theoretical properties (Corollary 2.7).