Thinnable Ideals and Invariance of Cluster Points

We define a class of so-called thinnable ideals 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 (xn) taking values in a separable metric space and a thinnable ideal I, it is shown that the set of I-cluster points of (xn) is equal to the set of 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 [15].