This is the answer Damien Gaboriau told me to the question from "my motivation" section. We can assume that the measure space is the interval $X=[0,1]$. Suppose we have a measured equivalence relation on $X$ whose equivalence classes are infinite countable. We want to show there exists a meaurable subset of $X$ of arbitrary small measure which intersects almost all equivalence classes.
Note that the map $x\mapsto I(x)=$ "infimum of the class of $x$" is measurable, so for almost all points $x$ the point $I(x)$ is not in the class of $x$, because otherwise we would have a measurable selector which is impossible. So assume for simplicity that $I(x)$ is never in the class of $x$. Then consider the family of sets $B_\epsilon$ for $\epsilon\in \mathbb R_+$. $B_\epsilon$ is the union $$ \bigcup_{x\in X} B(I(x), \epsilon)\cap E(x), $$ where $B(a,b)$ is the ball with center $a$ and radius $b$, and $E(x)$ is the equivalence class of $x$. The sets $B_\epsilon$ are a descending family with trivial intersection, so they have arbitrary small measures, but each of them intersects all classes.