$S(t_0, a)$ form a subset in $C([t_0, t_0 + T(t_0, a))\to X)$.

\[loc-fun\] A family $S(t_0, a)$, $t_0\in [0, +\infty)$, $a\in X$, will be called a family of abstract local integral funnels with terminal times $T(t_0, a)$ if they satisfy the following conditions.

1.  $T(t_0, a)$ is a lower semi-continuous function on $[0, +\infty)\times X$, i.e., if $(t_n, a_n)\to (t_0, a)$, then $T(t_0, a) \le \liminf T(t_n, a_n)$.

2.  Every set $S(t_0, a)$ is a non-empty compact in the space $C([t_0, t_0 + T(t_0, a))\to X)$ with the topology of uniform convergence on every closed subinterval $[\alpha, \beta]$ of $[t_0, t_0 + T(t_0, a))$. Every path $w(\cdot; t_0, a)\in S(t_0, a)$ is a continuous map from $[t_0, t_0 + T(t_0, a))$ into $X$, and $w(t_0; t_0, a) = a$.

3.  For every $t_0\ge 0$, the set-valued map $a\mapsto S(t_0, a)$ is measurable in the following sense. Each path $w$ in $S(t_0, a)$ can be re-parametrized as ${\tilde w}(s) = w(t_0 + s (T(t_0, a) - t_0))$ and then viewed as an element of the space ${\tilde \Omega} = C([0, 1)\to X)$. Denote by ${\tilde S}(t_0, a)$ the set $S(t_0, a)$ after such re-parametrization. We say that the map $a\mapsto S(t_0, 