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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 $|f(x)|<C$ and $C$ otherwise. Suppose we know that the Birkhoff averages $A_{N}f^{C}$ converge a.e. and in $L^1$ for each $C$. How do we conclude that the Birkhoff averages $A_{N}f$ converge a.e.? (I can see how to conclude they converge in $L^1$).

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(Not enough reputation to comment)

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.

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