2005/05/28 by Luis Baez-Duarte
Mathematics · #math.PR
published as Bull. Venezuela Acad. Sci. 1964 · 2 pages, Translation into English of a paper by the author generalizing Kac's theorem on the spatial mean of the Poincare cycle. Of possible pedagogical value
arxiv created 2005/05/28 · arxiv updated 2009/12/01
Let X be a measure space and T:X→ X a measurable transformation. For any measurable E⊆ X and x∈ E, the possibly infinite return time is nE(x):=inf\n>0: Tn x∈ E\. If T is an ergodic tranformation of the probability space X, and μ(E)>0, then a theorem of M. Kac states that ∫E nE dμ=1. We generalize this to any invertible measure preserving transformation T on a finite measure space X, by proving independently, and nearly trivially that for any measurable E⊆ X one has ∫E nE dμ=μ(IE), where IE is the smallest invariant set containing E. In particular this also provides a simpler proof of Poincaré's recurrence theorem.