X(\omega)$ for $\omega\in dom(u),$ where $\omega\in dom(u)$ is the domain of $u.$ Let $L$ be some set of sections.

[@G1]. A pair $(X, L)$ is said to be a [*measurable bundle of Banach spaces*]{} over $\Omega$ if

1.  $\lambda_1 c_1+\lambda_2 c_2\in L$ for all $\lambda_1, \lambda_2\in \mathbb{R}$ and $c_1, c_2\in L,$ where $\lambda_1 c_1+\lambda_2 c_2:\omega\in dom(c_1)\cap
       dom(c_2) \rightarrow \lambda_1 c_1(\omega)+\lambda_2 c_2(\omega);$

2.  the function $||c||:\omega\in  dom(c)\rightarrow
    ||c(\omega)||_{X(\omega)}$ is measurable for all $c\in L;$

3.  for every $\omega\in \Omega$ the set $\{c(\omega): c\in L, \omega\in dom(c)\}$ is dense in $X(\omega).$

A section $s$ is a step-section, if there are pairwise disjoint sets $A_1,A_2,\ldots,A_n\in\Sigma$ and sections $c_1,c_2,\ldots,c_n\in L$ such that $\bigcup\limits_{i=1}^n
 A_i=\Omega$ è $s(\omega)=\sum\limits_{i=1}^n
 \chi_{A_i}(\omega)c_i(\omega)$ for almost all $\omega\in\Omega$.

A section $u$ is measurable, if for any $A\in\Sigma$ there is a sequence $s_n$ of step-sections such that $s_n(\omega)\rightarrow u(\omega)$ for almost all $\omega\in A$.

Let $M(\Omega,X)$ be the set of all measurable sections. By symbol $L_0(\Omega,X)$ we denote factorization of the $M(\Omega,X)$ with respect to almost everywhere equality. Usually, by $\hat{u}$ we denote a class from $L_0(\Omega,X)$, containing a section $u\in M(\Omega,X)$, and by $\|\hat{u}\|$ we denote an element of $L_0(\Omega)$, containing $\|u(\omega)\|_{X(\omega)}$. Let ${\mathcal L^{\infty}}(\Omega,X)=\{u\in
M(\Omega,X):\|u(\omega)\|_{X(\omega)}\in \mathcal
L^{\infty}(\Omega)\}$ and $L^{\infty}(\Omega,X)=\{\widehat{u}\in
L_0(\Omega,X): 