compact subset of $\Omega^{t_0}_a$. Every path $u$ in $S(t_0, a)$ is parametrized as $u(t; t_0, a)$, where $t\ge t_0$.

2.  Each map $S(t_0, \cdot)$ is measurable, i.e., for every closed set $C\subset \Omega^{t_0}$, $$\{x\in X: S(t_0, x)\cap C \neq \emptyset\}\in {\cal B}_X\,.$$

3.  If $u\in S(t_0, a)$, then $\sigma_\tau u \in S(t_0+\tau, u(t_0+\tau))$.

4.  If $u\in S(t_0, a)$ and $v\in S(t_0+\tau, u(t_0+\tau))$, then the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, defined as in , belongs to the funnel $S(t_0, a)$.

\[thm:main\] Every family of abstract integral funnels $S(t_0, a)$, $t_0\ge 0$, $a\in X$, has, for every $t_0$, a measurable selection $ a\to u(\cdot; t_0, a)\in S(t_0, a)$ with the semigroup property . As a corollary, there is a Borel measurable (semi)process $U(t_1, t_0) : X\to X$ whose orbits are $U(t_1, t_0)(a) = u(t_l; t_0, a)$, for all $t_1 \ge t_0\ge 0$ and for all $a\in X$.

We modify our proof for the autonomous case from [@C-K]. The plan is to successively reduct each funnel $S(t_0, a)$ while preserving the properties [**S1 - S4**]{} and so that the limiting funnel would contain just one path; property [**S3**]{} for the limiting funnel then spells . The reduction of the funnels is 