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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 $a(1)<a(2)<\ldots$. 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? –  Ray Yang Feb 7 '13 at 12:21
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