Thinnable ideals and invariance of cluster points

Abstract

We define a class of so-called thinnable ideals $\mathcal {I}$ on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence $(x_n)$ taking values in a separable metric space and a thinnable ideal $\mathcal {I}$, it is shown that the set of $\mathcal {I}$-cluster points of $(x_n)$ is equal to the set of $\mathcal {I}$-cluster points of almost all of its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in Miller 1995 (MR1260176).