with respect to $\mu$ if and only if $$\frac{1}{N}\sum_{n=0}^{N-1} \langle f\circ T^n,g\rangle_{L^2(\mu)}\xrightarrow{N\to\infty} 0$$ for any $f,g\in L^2_0(\mu)$. The transformation $T$ is weakly mixing iff $$\frac{1}{N}\sum_{n=0}^{N-1} \left\vert \langle f\circ T^n,g\rangle_{L^2(\mu)}\right\vert\xrightarrow{N\to\infty} 0$$ for any $f,g\in L^2_0(\mu)$, and $T$ is strongly mixing iff $$\langle f\circ T^n,g\rangle_{L^2(\mu)}\xrightarrow{n\to\infty} 0\, .$$

Now, let us denote by $V_T:L^2(\mu)\to L^2(\mu)$ the Koopman operator associated with a measure-preserving transformation $T:(X,\mathfrak B,\mu)\to(X,\mathfrak B,\mu)$, the isometry defined by $$V_Tf=f\circ T\, .$$

For any $f,g\in L^2(\mu)$, there is a uniquely defined complex measure $\sigma_{f,g}=\sigma_{f,g}^T$ on $\TT$ such that $$\widehat\sigma_{f,g} (n)=\left\{
\begin{matrix} \langle V_T^n f,g\rangle_{L^2(\mu)}&{\rm if}&n\geq 0\\
\langle V_T^{*\vert n\vert} f,g\rangle_{L^2(\mu)}&{\rm if}&n< 0

\end{matrix}
\right.$$ (When $f=g$, this follows from Bochner’s theorem because in this case the right-hand side defines a positive-definite function on $\ZZ$; and then one can use a “polarization" argument). We denote by $\Sigma(T,\mu)$ the collection of all measures $\sigma_{f,g}$, $f,g\in L^2_0(\mu)$, and forgetting the measure $\mu$ we refer to $\Sigma(T,\mu)$ as “the spectral measure of $T$".

With these notations, we see that $T$ is weakly mixing with respect to $\mu$ iff all measures $\sigma\in \Sigma(T,\mu)$ are continuous (by Wiener’s lemma), and that $T$ is strongly mixing iff all measures $\sigma\in \Sigma(T,\mu)$ are Rajchman (by Rajchman’s lemma). Likewise, $T$ is 