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)
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.
Existence of a sequence which is good for mean convergence but not good for pointwise convergence
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analysis
ergodic-theory
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0I do not understand your question. Are you asking for a counterexample to a theorem which is known to be true? – 2013-02-07