The Fejer integrals and the von Neumann ergodic theorem with continuous time

The Fejer integrals for finite measures on the real line and the norms of the deviations from the limit in the von Neumann ergodic theorem both are calculating, in fact, with the same formulas (by integrating of the Fejer kernels) – and so, this ergodic theorem is a statement about the asymptotic of the growth of the Fejer integrals at zero point of the spectral measure of corresponding dynamical system. It gives a possibility to rework well-known estimates of the rates of convergence in the von Neumann ergodic theorem into the estimates of the Fejer integrals in the point for finite measures: for example, we obtain natural criteria of polynomial growth and polynomial decay of these integrals. And vice versa, numerous in the literature estimates of the deviations of Fejer integrals in the point allow to obtain new estimates of the rate of convergence in this ergodic theorem.