Kac's return-time formula for any measure-preserving map
A machine-checked proof of Kac's return-time formula. For a measure-preserving map f and a measurable set A, the integral over A of the first return time to A equals the measure of the points whose orbit ever meets A. The return time is valued in the extended naturals, infinite for a point that never returns, and with that convention the formula holds for every measure, with no finiteness, sigma-finiteness, conservativity or ergodicity assumption. For an ergodic map of a probability space and mu(A) > 0 the integral is 1, so the mean return time to A, starting in A, is 1 / mu(A).
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