Abstract
Given a space \(X\), a \(\sigma \)-algebra \(\mathfrak {B}\) on \(X\) and a measurable map \(T:X \rightarrow X\), we say that a measure \(\mu \) is half-invariant by \(T\) if, for any \(B \in \mathfrak {B}\), we have \(\mu \left( T^{-1}(B)\right) \le \mu (B)\). We will present a generalization of Birkhoff ergodic theorem to \(\sigma \)-finite half-invariant measures.
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Notes
Otherwise, \(\alpha <0\) and we may take \(-f\), \(-\alpha \) and \(-\beta \) instead.
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The authors were partially funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT—Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2013.
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de Carvalho, M.P., Moreira, F.J. A Note on the Ergodic Theorem. Qual. Theory Dyn. Syst. 13, 253–268 (2014). https://doi.org/10.1007/s12346-014-0116-x
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DOI: https://doi.org/10.1007/s12346-014-0116-x