Concluding the Birkhoff ergodic theorem from the corresponding statement for bounded functions.

Question

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 $L^1$).

Answer 1

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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.