I would like to know if there is a notion or an example related to the following situation: a transform $T$ on a space $E$, which is equipped with an infinite measure $\mu$, satisfies $\mu(A\cap T^{-n}B)\sim \mu(A)\mu(B)c\_n$ as $n$ tends to infinity, for any measurable subets $A$ and $B$ in some finite-measure subset of $E$, where $c\_n$ is regularly varying with an index between -1 and 0.


This may be related to the notion pointwise dual ergodicity in infinite ergodic theory （Aaronson 1997）.


Sorry for not being more precise since I'm very ignorant in this area. 



A regular variation in infinite ergodic theory
A theorem of Hajian, Ito and Kakutani shows that every infinite measure preserving transformation always has weakly wandering sets. That is a set $A$ and a subsequence $n\_k\to\infty$ such that $\left\{ T^{-n\_k}A\right\}\_{k=1}^\infty$ are pairwise disjoint. See Aaronsons book or the new book of Eigen, Hajian, Ito and Prasad. If you are willing to change the mixing condition of sets for decay of correlations of "nice" functions in smooth dynamical systems then the paper of Melbourne and Terhesiu <http://mat.univie.ac.at/~terhesiu/papers/invent_rev.pdf>
is a formulation similar to what you want.


