Product of probability-preserving systems

The product system of two probability-preserving systems $(X,\Sigma,\mu,T)$ and $(Y,\mathcal T,\nu,S)$ is

(X\times Y,\Sigma\otimes\mathcal T,\mu\otimes\nu,T\times S), \qquad(T\times S)(x,y)=(Tx,Sy).

One may complete the product measure. For group actions, the diagonal action uses the same $g$ on both coordinates.

Ergodicity of products

Two ergodic systems can have a nonergodic product: equal irrational rotations fix the difference of the two circle coordinates. The condition that $T\times T$ be ergodic is exactly weak mixing.