On the notion of reasonable functions in Cohen--Lenstra type heuristics

We elucidate the notion of “reasonable functions” in the context of Cohen–Lenstra heuristics. For a sequence $(x_i{)}_i$ of independent random variables valued in an infinite discrete probability space $(X,p)$, we investigate for what sets $S$ of functions on $X$ it is true that almost surely, for all $f\in S$, the average of $f$ on the sequence is equal to its expected value. We show that if $p$ decays sufficiently quickly, as e.g. the Cohen–Lenstra distribution does, then ${\bigcup }_{q>1}{L}^q(X)$ is such a set; while if $p$ decays sufficiently slowly, then not even ${\bigcap }_{q<\mathrm{\infty }}{L}^q(X)$ has this property.