denote an algebra of equivalence classes of essentially bounded measurable functions.

Let $\mathcal{U}$ be a linear space over the real field $\mathbb{R}$. By $\|\cdot\|$ we denote a $L_0(\Omega)$-valued norm on $U$. Then the pair $(\mathcal{U},\|\cdot\|)$ is called a [*lattice-normed space (LNS) over $L_0(\Omega)$*]{}. An LNS $\mathcal{U}$ is said to be [*$d$-decomposable*]{} if for every $x\in \mathcal{U}$ and the decomposition $\|x\|=f+g$ with $f$ and $g$ disjoint positive elements in $L_0(\Omega)$ there exist $y,z\in \mathcal{U}$ such that $x=y+z$ with $\|y\|=f$, $\|z\|=g$.

Suppose that $(\mathcal{U},\|\cdot\|)$ is an LNS over $L_0(\Omega)$. A net $\{x_\alpha\}$ of elements of $\mathcal{U}$ is said to be [*$(bo)$-converging*]{} to $x\in \mathcal{U}$ (in this case, we write $x=(bo)$-$\lim x_\alpha$), if the net $\{\|x_\alpha - x\|\}$ $(o)$-converges to zero in $L_0(\Omega)$ (written as $(o)$-$\lim \|x_\alpha -x\|=0$). A net $\{x_\alpha\}_{\alpha\in A}$ is called [*$(bo)$-fundamental*]{} if $(x_\alpha-x_\beta)_{(\alpha,\beta)\in A\times A}$ $(bo)$-converges to zero.

An LNS in which every $(bo)$-fundamental net $(bo)$-converges is called [*$(bo)$-complete*]{}. A [*Banach-Kantorovich space (BKS) over $L_0(\Omega)$*]{} is a $(bo)$-complete $d$-decomposable LNS over $L_0(\Omega)$. It is well known [@G1; @G2] that every BKS $\mathcal{U}$ over $L_0(\Omega)$ admits an $L_0(\Omega)$-module structure such that $\|fx\|=|f|\cdot\|x\|$ for every $x\in \mathcal{U},\ f\in L_0(\Omega)$, where $|f|$ is the modulus of a function $f\in L_0(\Omega)$.

A set 