I'm trying to prove an equivalent statement about weak mixing transformations that relies on the spectral theorem, but I can't find a reference to fill in the last details. A hint for solving it or finding details will be greatly appreciated.

The statement; a transformation $T:(X,\lambda) \rightarrow (X,\lambda)$ is weak mixing, i.e. $\frac{1}{N}\sum | \lambda(T^{-n}A \cap B) - \lambda(A)\lambda(B)|^2 \rightarrow 0$, if and only if the associated operator $U_T : L^2(X) \rightarrow L^2(X)$ has only constant eigenfunctions.

By the spectral theorem, if $\sigma$ is the spectrum of $U_T$, for any $f \in L^2(X)$ there is a measure $\mu \in \mathcal{M}(\sigma)$ for which $\int_\sigma z^n d\mu = < U_T^n f, f> $

What I'm missing; in the converse, if $U_T$ has only constant eigenvalues and $\int_X f = 0$ then the corresponding function $\mu$ is non atomic.

Thank you in advance. Diego