L_1(M,\Phi),$ in addition $|\widehat{\Phi}(x)|\leq
\widehat{\Phi}(|x|)$ [@CZ1]. For any $x\in L_1(M,\Phi),$ set $\|x\|_{1,\Phi}=\widehat{\Phi}(|x|).$ It is known that $L_1(M,\Phi)$ is a linear subspace of $S(M),$ $ML_1(M,\Phi)M
\subset L_1(M,\Phi),$ and $x^*\in L_1(M,\Phi)$ for all $x\in
L_1(M,\Phi)$ [@CZ1]. Let $\Phi$ be a faithful normal trace on $M$ with values in $L_0(\Omega)$ (associated with $Z$). Following [@CZ] for any $p > 1$, we set $L_p(M,\Phi) = \{x \in S(M) :
\|x\|_{p,\Phi} \in L_1(M,\Phi)\}$ and $\|x\|_{p,\Phi} =
\widehat{\Phi}(|x|^p)^{\frac{1}{p}}$.

[@CZ1; @CZ] \[1.1\] Let $x,y\in S(M)$. Then the following statements hold:

1.  $ \|x\|_{1,\Phi}=\sup\{|\Phi(xy)|: \|y\|_M\leq 1\}.$

2.  for $p>1$, one has $\|x\|_{p,\Phi}=\sup\{|\Phi(xy)|:
    \|y\|_{q,\Phi}\leq \mathbf{1}\},$ where $\frac{1}{p}+\frac{1}{q}=1.$

In the early 90’s of the last century A.E. Gutman [@G1] introduced the measurable Banach bundles with lifting axioms. It was established that every Banach–Kantorovich space over the ring of measurable functions can be expressed as measurable bundle of Banach spaces. Now let us recall some notions from this approach.

Let $X$ be a mapping which maps every point $\omega\in \Omega$ to some Banach space $(X(\omega),\|\cdot\|_{X(\omega)})$. In what follows, we assume that $X(\omega)\neq \{0\}$ for all $\omega\in \Omega.$ A function $u$ is said to be a *section* of $X$, if it is defined almost everywhere in $\Omega$ and takes its value $u(\omega)\in 