Let $G$ be an at most countable discrete group acting freely on a standard probability measure space $X$ in a measure preserving way.
It is well known that if $G$ is a finite group then this action admits a fundamental domain. As pointed out by Andreas below, by Rokhlin lemma, if $G$ contains an element of infinite order we can find an $(\varepsilon, N)$-fundamentalish domain $U$, where the latter is defined as follows:
Call a set $U\subset X$ an $(\varepsilon, N)$-fundamentalish domain iff there exist $N$ elements $g_1, \ldots, g_N$ of $G$ such that the sets $g_i(U)$ are pairwise disjoint and the measure of their union is at least $1-\varepsilon$.
Question: If $G$ is an infinite group, $N_0$ is a natural number, $\varepsilon_0$ is a positive real number, does there exist an $(\varepsilon, N)$-fundamentalish domain with $\varepsilon<\varepsilon_0$ and $N>N_0$?
For example when the action is profinite and "transitive on each level", then clearly answer is positive: there exist $(0,N)$-fundamentalish domains for arbitrary large $N$.