Infinitesimal containment and sparse factors of iid

We introduce infinitesimal weak containment for measure-preserving actions of a countable group $Γ$: an action $(X,μ)$ is infinitesimally contained in $(Y,ν)$ if the statistics of the action of $Γ$ on small measure subsets of $X$ can be approximated inside $Y$. We show that the Bernoulli shift $[0,1]^Γ$ is infinitesimally contained in the left-regular action of $Γ$. For exact groups, this implies that sparse factor-of-iid subsets of $Γ$ 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.