a)$ is measurable if, for any closed subset $F$ of ${\tilde \Omega}$ (with the compact-open topology), $$\{a\in X: {\tilde S}(t_0, a) \cap F \neq \emptyset\}\in {\cal B}_X\,.$$

4.  If $u\in S(t_0, a)$ and $\tau < T(t_0, a)$, then $T(t_0 + \tau, u(t_0 + \tau; t_0, a)) \ge 
    T(t_0, a) - \tau$ and $\sigma_\tau u \in S(t_0+\tau, u(t_0+\tau; t_0, a))$.

5.  If $u\in S(t_0, a)$, $\tau < T(t_0, a)$, and $v\in S(t_0+\tau, u(t_0+\tau; t_0, a))$, then the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, defined by analogy with , belongs to the funnel $S(t_0, a)$.

\[thm:2\] Every family of local abstract integral funnels $S(t_0, a)$, $t_0\in [0, +\infty)$, $a\in X$, with terminal times $T(t_0, a)$, has a selection ${\frak u}(\cdot; t_0, a)$ with the following properties.

1.  For every $t_0\ge 0$, the map $ X \owns a \mapsto {\frak u}(\cdot; t_0, a)\in C([t_0, t_0 + T(t_0, a))\to X)$ is measurable.

2.  ${\frak u}(t_0; t_0, a) = a$,

3.  ${\frak u}(t_2; t_1, {\frak u}(t_1; t_0, a)) = {\frak u}(t_2; t_0, a)$ for all $t_1\in [t_0, t_0 + T(t_0, a))$ and $t_2\in [t_1, t_0 + T(t_0, a))$.

We mimic the proof of Theorem \[thm:main\] with a few modifications. Let $\vp: X\to [0, 1]$ 