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The ergodic theorems talk about the limit behaviour of the ergodic averages. For example, if $(X,\chi,\mu,T)$ is a measure preserving system, where $\mu$ is a probability measure on $X$, then we have the mean ergodic theorem: $\text{If f}\in L^1(X),\text{ then we have that} \frac{1}{n}\sum_{i=1}^nf(T^ix) \text{ converges in } L^1.$ and the pointwise ergodic theorem: $\text{If f}\in L^1(X),\text{ then we have that } \frac{1}{n}\sum_{i=1}^nf(T^ix) \text{ converges almost everywhere.}$ Now we can consider a more general ergodic average of the form: $\frac{1}{n}\sum_{i=1}^nf(T^{a(i)}x)$ where $\{a(i)\}$ is a sequence of natural numbers with $a(1). If the averages converges in $L^1$, then we say that $\{a(i)\}$ is good for mean convergence, and similarly for pointwise convergence. In general not every sequence is good for the two kinds of convergence.

Now my question is that: does there exist a sequence which is good for mean convergence but not good for pointwise convergence? I think such one exists but can not give an example.

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    I do not understand your question. Are you asking for a counterexample to a theorem which is known to be true?2013-02-07

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