Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following statement is true: 



> 
> For any probability space
>  $(X,\Sigma,\mu),$ for any 
>  measure-preserving transformation $T$
>  on $X,$ for any $f\in L^1(X,\mu)$ the
>  ergodic means along $A,$
> 
> 
> $$M(f,T,A,t,x):=\big|\{j\in A\, : j\leq t\}\big|^{\,-1}\sum\_{j\in A,\, j\leq t}f(T^{j}x)$$
> 
> 
> converge for a.e. $x\in X$ to the conditional expectation w.r.to the $T$ invariant $\sigma$-algebra, $\mathbb{E}(f|\Sigma\_T),$ as $t \to +\infty.$
> 
> 
> 


So $\mathbb{N}$ itself is nice in this sense, by Birkhoff's theorem; if $A$ is nice and $m$ is a positive integer, the set of translates $A+m$ is nice (the set of convergence with $T$ along $A+m$ coinciding with the $T^{\, m}$ pre-image of the set of convergence with $T$ along $A$). Also, a disjoint union of two nice sets is nice.


Is there any other structure on the family of these sets? What about e.g. the union of two of them? (at a glance it seems to me that something more can be said for the analogous cases of other ergodic theorems, e.g. for the $L^p$ convergence.


Looking at this very related [question](https://mathoverflow.net/questions/42837/a-non-standard-ergodic-limit), and its answer, make me think that the situation may be non-trivial and interesting enough, so that it should have been studied. 



Ergodic limits along subsets of $\mathbb{N}.$
These are called good universal sets.
Bourgain (1987) proved that sequences of the form $p(n)$, $n \in {\bf N}$, $p$ a non constant polynomial, are good.


He also proved (1988) that the set of primes is a good universal set for $L^p$ functions, $p> {(1+\sqrt{3})\over 2}$. This was later improved to $p>1$ by Wierdl, see its article for a short introduction to the problem <http://www.springerlink.com/content/e027w4211n7784h1/fulltext.pdf> . There is now an extensive litterature on the problem, following Bourgain's articles. 


In another direction, note that a transformation T is mixing iff the ergodic theorem for T holds for all subsequences (see e.g. the book of Krengel, "ergodic theorems").


