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• W2059605173 abstract "Let (X, 6B, m) be a finite measure space. We shall denote by Lv(m) (1 ?p < oo) the Banach space of all real-valued 63-measurable functionsf defined on X such that if I P is m-integrable, and by Lo?(m) the Banach space of all real-valued, 63-measurable, m-essentially bounded functions defined on X; as usual, the norm in LP(m) is given by lIpIfI = {fxlf(x)IPdm}1/P, and the norm in Lcr(m) by IIg||OO -m-ess. supzEEx lg(x)l. Two functions in LP(m) or Lo((m) 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 L'(m) into L1(m) with III1Ti<1. We say that the pointwise ergodic theorem (the L'(m)-mean ergodic theorem, respectively) holds for such an operator T if for every f in L'(m), the sequence of averages { 1 (/n) k=0 Thf } converges m-almost everywhere (in the norm of L'(m), respectively) to a function in Ll(m). Recently, R. V. Chacon [1] constructed a class of positive linear operators in L1(m) with the norm equal to 1 for which the pointwise ergodic theorem fails to hold. Also, A. Ionescu Tulcea [5], [61 showed that in the group of all positive invertible linear isometries of L(r(m) 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(r(m) into Lco (m) and IIT| < 1, 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 ITI1 <1, then the pointwise ergodic theorem is always valid. This is because, for such an operator T, nI Tnf(x) I < rnm-almost everywhere for every f in L' (m), since" @default.
• W2059605173 created "2016-06-24" @default.
• W2059605173 creator A5055085996 @default.
• W2059605173 date "1965-01-01" @default.
• W2059605173 modified "2023-09-27" @default.
• W2059605173 title "Uniform integrability and the pointwise ergodic theorem" @default.
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• W2059605173 doi "https://doi.org/10.1090/s0002-9939-1965-0171895-3" @default.
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