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A general local ergodic theorem in \(L_ 1\). (English) Zbl 0579.47004

Let \(\{T_ t\}_{t>0}\) be a strongly continuous semi group of linear contractions on the \(L_ 1\) space of a measure space (X,\({\mathcal F},\mu)\). It is shown that if \(f\in L_ 1\) then \(\lim_{t\to 0^+}(1/t)\int^{t}_{0}T_ sf ds\) exists a.e. on X.

MSC:

47A35 Ergodic theory of linear operators
47D07 Markov semigroups and applications to diffusion processes
28D05 Measure-preserving transformations
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