An Extension of Chacon-Ornstein Ergodic Theorem

Abstract

Let (X,X,μ) be a σ-finite measure space. If P is an L¹(X,X,μ) positive linear contraction (i.e. Pf≥0 for fεL¹, f≥0 and ∫|Pf|d03BC;≤ ≤ ∫|f|dμ for fεL¹) the celebrated theorem of Chacon and Ornstein ([3]) asserts that for f,gεL¹, g≥0:

$$\lim_{n\rightarrow \infty }\frac{f+pf+...+p^{n}f}{g+pg+...p^{n}g}$$

exists and is finite μ-a.e. on the set $\left \{x|\sum_{i=0}^{\infty }(p^{i}g)(x)> 0 \right \}$. In [2] Chacon gives an extension of this result for nonpositive contractions: if T is an arbitrary L¹-contraction, fεL¹ and {p_n}_nεN is a sequence of positive measurable functions such that for every nεN, gεL¹ ₊, g≤P_n implies |Tg|≤p_n+1, then

$$\lim_{n\rightarrow \infty }\frac{f+Tf+...+T^{n}f}{p_{0}+p_{1}+...+p_{n}}$$

exists and is finite μ-a.e. on the set $\left \{x|\sum_{i=0}^{\infty }(p_{i})(x)> 0 \right \}$.