I am master student.I have been starting to learn Ergodic theory.
Let $(X,A,\mu,T)$ be a measure-preserving dynamical system.
Birkhoff Ergodic theorem
$$\lim_{n\to\infty} \frac{1}{n}\sum_{i=0}^{n-1}g(T^{i}(x))$$ (is called Birkhoff average of $g$) for every $g\in L^{1}(\mu)$ converges a.e.(every where if $g$ is continuous)
I have the following question:
Do we have thereom like Birkhoff ergodic theorem such that it say Birkhoff sum converges every where?
Assume that the system $(X,A,\mu,T)$ is ergodic. The a. e. convergence of the Birkhoff sums would require $\int g\, d\mu = 0$. But, even under this assumption we do not need to have convergence a.e.
Namely, Ulrich Krengel in his 1978 paper On the speed of convergence in the ergodic theorem proved that for any ergodic measure preserving invertible transformation $T$ on $[0,1]$ with Lebesgue measure $\lambda$ and any positive sequence $\alpha_n \to 0$ as $n \to \infty$ one can find (even a continuous) function $g$, with $\int g\, d\lambda = 0$, and such that $$ \limsup_{n\to\infty} \frac{1}{\alpha_n} \biggl\lvert \frac{1}{n} \sum\limits_{i=1}^{n-1} g(T^i(x)) \biggr\rvert = \infty $$ almost everywhere. Taking $\alpha_n=1/n$ we have $$ \limsup_{n\to\infty}\, \biggl\lvert \sum\limits_{i=1}^{n-1} g(T^i(x)) \biggr\rvert = \infty $$ almost everywhere.