Bernoulli shift - Knowlpedia

For a finite alphabet $A$ with probability distribution $p$ , the two-sided Bernoulli shift is

X=A^{\mathbb Z},\qquad \mu=p^{\mathbb Z},\qquad (Tx)_j=x_{j+1}.

The product sigma-algebra is generated by events depending on finitely many coordinates. The shift is invertible and measure preserving, and is strongly mixing.

Why it mixes

Two events depending on finite coordinate sets become independent when a sufficiently large shift separates those sets. Hence their correlation then equals the product of their probabilities exactly. Approximation by these finite-coordinate events proves mixing for all measurable events.

One-sided version

On $A^{\mathbb N_0}$ , the same formula drops the initial coordinate. Unless $p$ is concentrated at one symbol, this version is noninvertible; its forward Koopman operator remains an isometry. Binary digits identify the fair one-sided binary shift with the doubling map modulo the usual null set of ambiguous expansions.

References

Jesse Peterson, Lecture notes on ergodic theory, 2011. §2.1, Bernoulli actions.