ergodic iff $\sigma (\{ \mathbf 1\})=0$ for every $\sigma\in \Sigma(T,\mu)$.

More generally, given any family of measures $\mathcal B\subset\mathcal M(\TT)$, one may say that $T$ is *$\mathcal B$-mixing* with respect to $\mu$ if the spectral measure of $T$ lies in $\mathcal B$, all measures $\sigma\in\Sigma(T,\mu)$ are in $\mathcal B$. We shall in fact consider a more specific case which seems to be the most natural one for our concerns. Let us denote by $\mathcal F_+:\mathcal M(\TT)\to \ell^\infty(\ZZ_+)$ the positive part of the Fourier transformation, i.e. $\mathcal F_+(\sigma)= \widehat\sigma_{\vert \ZZ_+}$.

Given any family $\mathbf S\subset\ell^\infty(\ZZ_+)$, we say that a measure $\sigma\in\mathcal M(\TT)$ is *$\mathbf S$-continuous* if $\mathcal F_+(\sigma)\in\mathbf S$. A measure-preserving transformation $T:(X,\mu)\to (X,\mu)$ is *$\mathbf S$-mixing* with respect to $\mu$ if every measure $\sigma \in\Sigma (T,\mu)$ is $\mathbf S$-continuous.

Thus, strong mixing is just $\mathbf S$-mixing for the family $\mathbf S=c_0(\ZZ_+)$, weak mixing is $\mathbf S$-mixing for the the family $\mathbf S$ of all sequences $(a_n)\in\ell^\infty(\ZZ_+)$ such that $\vert a_n\vert\to 0$ in the Cesàro sense, and ergodicity corresponds to the family $\mathbf S$ of all $a\in\ell^\infty (\ZZ_+)$ tending to $0$ in the Cesàro sense. In what follows, these families will be denoted by $\mathbf S_{\rm strong}$, $\mathbf S_{\rm weak}$ and 