For an ergodic measure-preserving T on (X, μ) and a set A with μ(A) > 0, the first-return time τ_A(x) = inf{n ≥ 1 : T^n x ∈ A} satisfies ∫_A τ_A dμ = μ(∪_n T^n A) = 1; hence E[τ_A | A] = 1/μ(A). Kac 1947. Foundation of the…
For an ergodic measure-preserving T on (X, μ) and a set A with μ(A) > 0, the first-return time τ_A(x) = inf{n ≥ 1 : T^n x ∈ A} satisfies ∫_A τ_A dμ = μ(∪_n T^n A) = 1; hence E[τ_A | A] = 1/μ(A). Kac 1947. Foundation of the…