  In this paper we prove a general ergodic theorem for ergodic and measure
preserving actions of R^d on standard Borel spaces. In particular, we cover
R.L. Jones ergodic theorem on spheres. Our main theorem is concerned with
ergodic averages with respect to homogeneous dilations of Rajchman measures on
Rd . We establish mean convergence in Hilbert spaces for general Rajchman
measures, and give a criterion in terms of the Fourier dimension of the measure
when almost everywhere pointwise convergence holds. Applications include
averages over smooth submanifolds and polynomial curves.
