  We prove that every hyperbolic measure invariant under a C^{1+\alpha}
diffeomorphism of a smooth Riemannian manifold possesses asymptotically
``almost'' local product structure, i.e., its density can be approximated by
the product of the densities on stable and unstable manifolds up to small
exponentials. This has not been known even for measures supported on locally
maximal hyperbolic sets.
  Using this property of hyperbolic measures we prove the long-standing
Eckmann-Ruelle conjecture in dimension theory of smooth dynamical systems: the
pointwise dimension of every hyperbolic measure invariant under a C^{1+\alpha}
diffeomorphism exists almost everywhere. This implies the crucial fact that
virtually all the characteristics of dimension type of the measure (including
the Hausdorff dimension, box dimension, and information dimension) coincide.
This provides the rigorous mathematical justification of the concept of fractal
dimension for hyperbolic measures.
