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  "body": "\n## Statement\n\nLet $(X, \\mathcal{B}, \\mu)$ be a [[probability space|0NFPJVJJHX]] and let $T \\colon X \\to X$ be an [[ergodic|AZRPHS478B]] [[measure-preserving transformation|556WBDAHNP]].\nLet $A \\in \\mathcal{B}$ with $\\mu(A) \u003e 0$, and for $x \\in A$ let\n$$r_A(x) = \\min\\set{n \\ge 1 : T^n x \\in A}$$\nbe the first return time of $x$ to $A$, which is finite for almost every $x \\in A$.\nThen\n$$\\int_A r_A \\, d\\mu = 1, \\qquad \\text{that is,} \\qquad \\frac{1}{\\mu(A)} \\int_A r_A \\, d\\mu = \\frac{1}{\\mu(A)}.$$\n\n## Remarks\n\nThe return time is finite almost everywhere on $A$ by the [[Poincaré recurrence theorem|TVYJPBHD0S]].\nFor invertible $T$, let $A_n = \\set{x \\in A : r_A(x) = n}$.\nThe sets $T^j A_n$ with $n \\ge 1$ and $0 \\le j \u003c n$ are pairwise disjoint, and their union is invariant up to a null set and contains $A$, so by ergodicity it has measure $1$.\nHence $1 = \\sum_{n \\ge 1} n \\, \\mu(A_n) = \\int_A r_A \\, d\\mu$.\nThe non-invertible case follows by passing to the natural extension.\nExample: for the rotation $x \\mapsto x + 1/q \\bmod 1$ on $\\set{0, 1/q, \\dots, (q-1)/q}$ with the uniform measure and $A$ a single point, the return time is $q$ and $\\int_A r_A \\, d\\mu = q \\cdot \\frac{1}{q} = 1$.\nWithout ergodicity the integral equals the measure of the set of points whose orbit meets $A$, which can be less than $1$.\nFor the identity map, $r_A = 1$ on $A$ and $\\int_A r_A \\, d\\mu = \\mu(A)$.\n\n## Explanations\n\n### L1 human\n\nThe rarer a set of states, the longer it takes to come back to it, and the lemma gives the exact rate.\nIf a set has probability $\\mu(A)$, then starting from that set, the average waiting time to return is $1/\\mu(A)$.\nFor example, if a fair die is rolled again and again, a six comes back after $6$ rolls on average.\n\n### L2 human\n\nKac's lemma explains why Poincaré recurrence does not contradict the second law of thermodynamics.\nFor a macroscopic system, the set of states near a given low-entropy state has astronomically small measure.\nSo the mean return time is astronomically long.\nIt is also the basis of the induced transformation $T_A(x) = T^{r_A(x)} x$ on $A$, which preserves the normalized measure on $A$ and is ergodic when $T$ is.\nFor an ergodic Markov chain started at a state $s$ with stationary probability $\\pi(s) \u003e 0$, it gives the classical formula $1/\\pi(s)$ for the expected return time to $s$.\n",
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