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  "body": "\n## Example\n\nLet $A = \\set{1, \\dots, k}$ be a finite alphabet and let $p = (p_1, \\dots, p_k)$ be a probability vector, with $p_i \\ge 0$ and $p_1 + \\cdots + p_k = 1$.\nLet $X = A^{\\Z}$ be the set of two-sided sequences $x = (x_n)_{n \\in \\Z}$, with the σ-algebra generated by the cylinder sets\n$$[a_m, \\dots, a_n]_m = \\set{x \\in X : x_i = a_i \\text{ for } m \\le i \\le n}.$$\nLet $\\mu$ be the [[infinite product measure|2HWVFHM13R]] of copies of $p$, so $\\mu([a_m, \\dots, a_n]_m) = p_{a_m} p_{a_{m+1}} \\cdots p_{a_n}$.\nThe shift $\\sigma \\colon X \\to X$, $(\\sigma x)_n = x_{n+1}$, is the *Bernoulli scheme* or *Bernoulli shift* $B(p_1, \\dots, p_k)$.\n\nCheck that $\\sigma$ is a [[measure-preserving transformation|556WBDAHNP]]: $x \\in \\sigma^{-1}[a_m, \\dots, a_n]_m$ means $x_{i+1} = a_i$ for $m \\le i \\le n$.\nSo $\\sigma^{-1}[a_m, \\dots, a_n]_m = [a_m, \\dots, a_n]_{m+1}$, which has the same measure $p_{a_m} \\cdots p_{a_n}$.\nCylinder sets form a π-system generating the σ-algebra, so $\\mu(\\sigma^{-1} E) = \\mu(E)$ for every measurable $E$.\n\nThe coordinate functions $x \\mapsto x_n$ form an i.i.d. sequence with law $p$, a two-sided version of the [[Bernoulli process|0CQQG3DEW9]] when $k = 2$.\nThe shift is invertible, [[mixing|S3RJQ9CX2P]], and a [[Kolmogorov automorphism|YRHJA7C7HS]].\nIts [[Kolmogorov–Sinai entropy|X1D1C7VH6P]] is $-\\sum_{i=1}^{k} p_i \\log p_i$, computed with the generating partition into the sets $\\set{x : x_0 = i}$.\nThe one-sided version uses $A^{\\N}$ instead of $A^{\\Z}$ and gives a non-invertible measure-preserving map.\n\n## Explanations\n\n### L1 human\n\nA Bernoulli scheme is the model of rolling the same loaded die once every second, forever into the past and into the future.\nA point of the space is a full record of all the rolls, and the shift moves the clock forward by one second.\nBecause the rolls are independent and identically distributed, moving the clock does not change the probabilities, so the shift preserves the measure.\nBernoulli schemes are the most random measure-preserving systems and serve as the yardstick for the others.\n",
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