The local ergodic theorem for groups of unitary operators and second order stationary processes

Let $(U_t)^\infty_{-\infty}$, be a strongly continuous unitary group in $L_2(X,S,\mu)$, where $\mu$ is a $\sigma$-finite measure.
The local ergodic theorem is the relation
\begin{equation} \lim_{t\to0}\frac1t\int^t_0(U_\tau f)(x)\,d\tau=f(x)\quad \text{a.\,e.} \end{equation}
for $f\in L_2(X)$. It is shown that this relation is not satisfied for all $f\in L_2(X)$ and $\{U_t\}$. Necessary and sufficient conditions are obtained for the local ergodic theorem in terms of properties of the spectral measure $\{E(d\lambda)f\}$, where $\{E(d\lambda)\}$ is the resolution of the identity corresponding to the group $(U_t)$. In particular, (1) is satisfied if the integral
$$ \int^\infty_{-\infty}[\log\log(\lambda^2+2)]^2\cdot\|E(d\lambda)f\|^2 $$
converges. Generalizations to multiparameter groups and homogeneous random fields are given.
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