BibTeX
@MISC{Avila_onthe,
author = {Artur Avila and Jairo Bochi},
title = {ON THE SUBADDITIVE ERGODIC THEOREM},
year = {}
}
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Abstract
Abstract. We present a simple proof of Kingman’s Subadditive Ergodic The-orem that does not rely on Birkhoff’s (Additive) Ergodic Theorem and there-fore yields it as a corollary. 1. Statements Throughout this note, let (X,A, µ) be a fixed probability space and T: X → X be a fixed measurable map that preserves the measure µ. Birkhoff’s Ergodic Theorem ([B]). Let f1: X → R be an integrable function, and let (1) fn = n−1∑ j=0 f1 ◦ T j for all n ≥ 1. Then fn/n converges a.e. to an integrable function f such that f = f1. Kingman’s Subadditive Ergodic Theorem ([Ki]). Let fn: X → R be a se-quence of measurable functions such that f+1 is integrable and (2) fm+n ≤ fm + fn ◦ Tm for all m, n ≥ 1. Then fn/n converges a.e. to a function f: X → R. Moreover, f+ is integrable and∫ f = lim n→∞ 1 n
