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.
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.
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.