  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 its subsequences, in the sense of
Lebesgue measure. Lastly, we obtain a characterization of ideal convergence,
which improves the main result in [Trans. Amer. Math. Soc. 347 (1995),
1811--1819].
