the measured groupoid has a Følner property. This is a key for the proof of the main theorem. In the last section, we make comments on uniform local amenability (ULA) defined by Brodzki, Niblo, Špakula, Willett, and Wright [@BNSWW Definition 2.2]. Definition of weak expander sequence is related to ULA.

Generalized box space and weak expander sequence {#SectionGenBox}
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Coarse space
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We prepare several notations related to coarse geometry. See Roe’s lecture note [@RoeLectureNote Chapter 2] for details. Let $X$ be a set. For subsets $T, T_1, T_2 \subseteq X^2$, we define the inverse $T^{-1}$ and the product $T_1 \circ T_2$ as follows: $$\begin{aligned}
T^{-1} &=& \{(x, y) \in X^2 \ |\ (y,x) \in T\},\\
T_1 \circ T_2 &=& \{ (x, y) \in X^2 \ 
|\ \textrm{there exists\ } z \in X \textrm{\ such that\ }
(x, z) \in T_1, (z, y) \in T_2 \}.\end{aligned}$$ Denote by $T^{\circ n}$ the $n$-th power $T \circ T \circ \cdots \circ T$. For a subset $Y \subseteq X$ and $T \subseteq X^2$, let $T[Y]$ be a set defined by $$\begin{aligned}
T[Y] = \{x \in X\ | \ \textrm{there exists\ } y \in Y \textrm{\ such that\ } (x, y) \in T \}.\end{aligned}$$ 