Mixing correlations extend to L2 functions

The constant one has $L^2$ norm one. Complex Cauchy–Schwarz gives $\mathrm{\mid }\int f\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_2$ and $\mathrm{\mid }\int g\mathrm{\mid }\le \mathrm{\parallel }g{\mathrm{\parallel }}_2$. Koopman isometry, iterated n times, gives $\mathrm{\parallel }f\circ T^n{\mathrm{\parallel }}_2=\mathrm{\parallel }f{\mathrm{\parallel }}_2$. Applying the real-modulus Cauchy–Schwarz inequality to $\mathrm{\mid }f\circ T^n\mathrm{\mid }$ and $\mathrm{\mid }g\mathrm{\mid }$ proves absolute integrability of the product. Thus every term defining $C_n$ exists and is independent of representatives, with the indicated complex pairing.

For indicators $C_n({\mathbf{1}}_B,{\mathbf{1}}_A)=\mu (A\cap T^{-n}B)-\mu (A)\mu (B)$. For finite complex simple $a={\sum }_j{\alpha }_j{\mathbf{1}}_{B_j}$ and $b={\sum }_k{\beta }_k{\mathbf{1}}_{A_k}$, sesquilinearity gives $C_n(a,b)={\sum }_{j,k}{\alpha }_j\overline{{\beta }_k}{C}_n({\mathbf{1}}_{B_j},{\mathbf{1}}_{A_k})$. Therefore the set version of strong mixing gives convergence for simple pairs, and the weak version gives it for absolute Cesaro averages by the finite triangle inequality.

For any $L^2$ h,k, the two Cauchy–Schwarz estimates in step 1.1 give $\mathrm{\mid }{C}_n(h,k)\mathrm{\mid }\le 2\mathrm{\parallel }h{\mathrm{\parallel }}_2\mathrm{\parallel }k{\mathrm{\parallel }}_2$. Choose finite simple a,b with $\mathrm{\parallel }f-a{\mathrm{\parallel }}_2<\delta$ and $\mathrm{\parallel }g-b{\mathrm{\parallel }}_2<\delta$ using only the finite-simple clause of density. Then sesquilinearity gives $\mathrm{\mid }{C}_n(f,g)-C_n(a,b)\mathrm{\mid }\le 2(\mathrm{\parallel }f-a{\mathrm{\parallel }}_2\mathrm{\parallel }g{\mathrm{\parallel }}_2+\mathrm{\parallel }a{\mathrm{\parallel }}_2\mathrm{\parallel }g-b{\mathrm{\parallel }}_2)\le 2\delta (\mathrm{\parallel }g{\mathrm{\parallel }}_2+\mathrm{\parallel }f{\mathrm{\parallel }}_2+\delta )\mathrm{.}$ The bound is independent of n.

Taking limsups of absolute values, or of their Cesaro averages, uses step 2.1 to remove the simple-pair term. Sending $\delta$ to zero proves the respective $L^2$ property. Conversely the $L^2$ property applies to indicators, which lie in $L^2$ because $\mu (X)=1$, and their identity in step 2.1 is precisely the set property.