Vol. 39, No. 3, 1971

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On the ratio ergodic theorem for semi-groups

Humphrey Sek-Ching Fong and Louis Sucheston

Vol. 39 (1971), No. 3, 659–667

For a semi-group Γ of positive linear contractions on L1 of a σ-finite measure space (X,𝒜), strongly continuous on (0,), there are two ratio ergodic theorems: one, due to Chacon and Ornstein, describes the behavior at infinity; the other one, due to Krengel-Ornstein-Akcoglu-Chacon, describes the “local” behavior. In the present paper we attempt to generalize these results to the case when the semigroup is only uniformly bounded. Then the space X decomposes into two parts, Y and Z, called the remaining and the disappearing part, and both ratio theorems are shown to hold on Y . The ratio theorem at infinity fails on Z.

Mathematical Subject Classification
Primary: 47D05
Secondary: 28A65
Received: 26 January 1970
Published: 1 December 1971
Humphrey Sek-Ching Fong
Louis Sucheston