Uniform Integrability and the Pointwtse Ergodic Theorem

Let iX, 03, m) be a finite measure space. We shall denote by L*im) (1 èp< °°) the Banach space of all real-valued (B-measurable functions/ defined on X such that |/[ p is m-integrable, and by L°°(w) the Banach space of all real-valued, (B-measurable, w-essentially bounded functions defined on X; as usual, the norm in Lpim) is given by 11/11,.= {fx\fix)\pdm}1'*, and the norm in Lxim) by \\g\\x = w-ess. suplSx |g(#)|Two functions in Lpim) or LKim) will be identified if they differ only on a set of m-measure zero. In this note, we shall be concerned with a positive linear operator T of Lxim) into L\m) with ||r||i^l. We say that the pointwise ergodic theorem ithe L\m)-mean ergodic theorem, respectively) holds for such an operator T if for every / in L^w), the sequence of averages {1 (/«) ]>^t-o Tkf} converges w-almost everywhere (in the norm of Llim), respectively) to a function in Llim). Recently, R. V. Chacon [l] constructed a class of positive linear operators in Lxiyn) with the norm equal to 1 for which the pointwise ergodic theorem fails to hold. Also, A. Ionescu Tulcea [5], [ô] showed that in the group of all positive invertible linear isometries of Lxim) the set of all T for which the pointwise ergodic theorem fails to hold forms a set of second category with respect to the strong operator topology. On the other hand, the ergodic theorem of HopfDunford-Schwartz [4] tells us that if, in addition, T maps L°°im) into LKim) and ||r||«,:Sl, then the pointwise ergodic theorem is valid for such T. In view of these facts, it is interesting to find out what other additional conditions on T would guarantee the validity of the pointwise ergodic theorem. In this note, we shall find a few such conditions which are weaker than the condition of the Hopf-Dunford-Schwartz theorem (though our conditions seem to work for a finite measure space only). We also obtain a result (corollary to Theorem 1, below) which generalizes a result obtained by N. Dunford and D. S. Miller in [3]. First of all, let us observe that if our operator T satisfies ||r||i<l, then the pointwise ergodic theorem is always valid. This is because, for such an operator T, 2"-o I Tnfix) \ < » m-almost everywhere for every /in Lxim), since