Many proofs of the Birkhoff ergodic theorem for probability measure preserving transformations first show that the Birkhoff averages converge for essentially bounded functions $f$, and then extend to general integrable functions by a truncation procedure.
I am confused about this last step.
Suppose $f\in L^{1}$ and let $f^{C}(x)=f$ if $|f(x)|
Concluding the Birkhoff ergodic theorem from the corresponding statement for bounded functions.
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functional-analysis
ergodic-theory
1 Answers
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I think the reduction to bounded functions is only made for proving $L^1$-convergence. The proofs of a.e. convergence normally don't need any boundedness assumption on $f \in L^1$. I checked in several standard references (Halmos, Petersen, Katok-Hasselblatt and more) and they all prove a.e. convergence directly.