Let $U$ be the koopman operator on $L^2(X,\mu)$ where $(X,T,\mu)$ is an MPT. We characterize the spectral measure by its fourier coefficients using bochner's theorem as $$\hat{\sigma_{f,g}}(-k) = \langle U^k f, g\rangle = \int_0^1 e^{2\pi i k \omega} \,\mathrm d\sigma_{f,g}(\omega)\quad \text{for} \quad f,g \in L^2(K,\mu). $$ Can someone provide me the details for why $$ \sigma_{f,g} = \text{weak}^*-\lim\limits_{N\rightarrow \infty} \frac{1}{N} \left\langle \sum_{n=0}^{N-1} e^{-2\pi i n \omega} U^n f, \sum_{n=0}^{N-1} e^{-2\pi i n \omega} U^n g \right\rangle \mathrm d \omega $$
The proof strategy is to show that the fourier coefficients of the measures on the RHS converge to $\hat{\sigma}_{f,g}(k)\ \forall k \in \mathbb{Z}$.