Infinitesimal containment and sparse factors of iid

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $\mathrm{\Gamma }$: an action $(X,\mu )$ is infinitesimally contained in $(Y,\nu )$ if the statistics of the action of $\mathrm{\Gamma }$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1{]}^{\mathrm{\Gamma }}$ is infinitesimally contained in the left-regular action of $\mathrm{\Gamma }$. For exact groups, this implies that sparse factor-of-iid subsets of $\mathrm{\Gamma }$ are approximately hyperfinite. We use it to quantify a theorem of Chifan–Ioana on measured subrelations of the Bernoulli shift of an exact group. For the proof of infinitesimal containment we define \emph{entropy support maps}, which take a small subset $U$ of $\{0,1{\}}^I$ and assign weights to coordinates above every point of $U$, according to how ‘‘important’’ they are for the structure of the set.