Hopf's ratio ergodic theorem by inducing

Abstract

We present a very quick and easy proof of the classical StepanovHopf ratio ergodic theorem, deriving it from Birkho¤’s ergodic theorem by a simple inducing argument. During the last few years, there has been some interest in short and easy proofs of (pointwise) ergodic theorems, naturally focussing on the most fundamental one, i.e. on Birkho¤’s result for probability preserving transformations, see e.g. [KW], [Ke], [P], and [Sh]. In [KK] a similar proof of an important extension was given, which came shortly after the discovery of the rst ergodic theorems ([N] and [B], see [Z] for historical comments): the Stepanov-Hopf ratio ergodic theorem ([St], [H]) which is the proper version of the pointwise ergodic theorem for in nite measure preserving transformations (there is no way to get a.e. convergence for ergodic sums normalized by a sequence of constants, cf. [A], §2.4). The aim of the present note is to point out that this result can also be derived as a direct consequence of Birkho¤’s theorem, via a (very) simple inducing argument (which doesn’t seem to be available or hinted at in the literature I know). We are going to prove Theorem 1 (Hopf’s Ratio Ergodic Theorem). Let T be a measure preserving transformation on the nite measure space (X;A; ). Let f; g 2 L1( ) with g 0 and R X g d > 0. Then there exists a measurable function Q(f; g) : X ! R such that Sn(f) Sn(g) = Pn 1 k=0 f T k Pn 1 k=0 g T k ! Q(f; g) a.e. on fsupn Sn(g) > 0g as n!1. On the conservative part the limit function Q(f; g) is measurable w.r.t. the algebra I A of T -invariant sets and satis es R I Q(f; g) g d = R I f d for all I 2 I. In particular, if g > 0 a.e., then Q(f; g) = E g f g kI , where d g := g d , and if T is ergodic, then Q(f; g) = R X f d = R X g d a.e. 2000 Mathematics Subject Classi cation. Primary 28D05, 37A30.