Weak ergodic averages over dilated measures

Let m is an element of N and X = (X, X, mu, (T-alpha)(alpha Rm)) be a measure-preserving system with an R-m-action. We say that a Borel measure nu on R-m is weakly equidistributed for X if there exists A subset of R of density 1 such that, for all f is an element of L-infinity (mu), we have lim t is an element of A,t ->infinity integral(Rm) f(Tt(alpha)x) d nu(alpha) = integral(x) f d mu for mu-almost every x 2 X. Let W.X / denote the collection of all ff 2 Rm such that the R-action (T-t alpha)(t is an element of R) is not ergodic. Under the assumption of the pointwise convergence of the double Birkhoff ergodic average, we show that a Borel measure nu on R-m is weakly equidistributed for an ergodic system X if and only if nu (W(X) + beta) = 0 for every beta is an element of R-m. Under the same assumption, we also show that nu is weakly equidistributed for all ergodic measure-preserving systems with R-m-actions if and only if nu(l) = 0 for all hyperplanes of R-.(m) Unlike many equidistribution results in literature whose proofs use methods from harmonic analysis, our results adopt a purely ergodic-theoretic approach.