Mixing is checkable on a generating pi-system

Let $\mathcal{C}$ consist of finite Boolean combinations of members of $\mathcal{P}$. Each indicator of an atom of a finite Boolean partition is a product of factors ${\mathbf{1}}_P$ and $1-{\mathbf{1}}_P$. Expanding the product expresses it as a finite integer linear combination of indicators of intersections of members of $\mathcal{P}$. Empty intersections are X, which lies in $\mathcal{P}$, and all other intersections lie in $\mathcal{P}$ by the pi-system property. Finite sums of these atom indicators express every ${\mathbf{1}}_C$, $C\in \mathcal{C}$, in this way.

Integration and multiplication of finite sums now express $d_n(C,D)$ as a finite linear combination ${\sum }_{j,k}{a}_j{b}_k{d}_n(P_j,Q_k)$. In the strong case each summand tends to zero. In the weak case the average of the absolute value is at most ${\sum }_{j,k}\mathrm{\mid }{a}_j{b}_k\mathrm{\mid }{N}^{-1}{\sum }_{n<N}\mathrm{\mid }{d}_n(P_j,Q_k)\mathrm{\mid }$, which tends to zero. The respective test therefore holds on $\mathcal{C}$.

The family $\mathcal{C}$ is an algebra generating $\mathcal{A}$. For measurable A,B and $\delta >0$, choose C,D in it with $\mu (A\mathrm{△}C),\mu (B\mathrm{△}D)<\delta$. Measure preservation gives $\mu (T^{-n}(B\mathrm{△}D))=\mu (B\mathrm{△}D)$ by repeated pullback. Thus the difference of the intersection measures is bounded by the sum of these two errors. Also $\mathrm{\mid }\mu (A)\mu (B)-\mu (C)\mu (D)\mathrm{\mid }\le \mu (A\mathrm{△}C)+\mu (B\mathrm{△}D)$, since all masses are at most one. Consequently $\mathrm{\mid }{d}_n(A,B)-d_n(C,D)\mathrm{\mid }<4\delta$ uniformly in n.

In the strong case the limsup of $\mathrm{\mid }{d}_n(A,B)\mathrm{\mid }$ is at most $4\delta$. In the weak case the same bound holds for the limsup of its absolute Cesaro averages. Letting $\delta$ decrease to zero proves the full respective property. Conversely either full property restricts to the pairs in $\mathcal{P}$ by its definition.