is *measurable* if $\Gamma^{-}(F)\in {\cal B}_X$ for every closed set $F\subset\Omega$. Since $\Omega$ is metric, if $\Gamma$ is measurable, it is weakly measurable, [@A-B Lemma 18.2]. Since $\Omega$ is in addition separable, if $\Gamma$ is *compact-valued* and weakly measurable, it is measurable, [@A-B Theorem 18.10]. Thus, in our setting, for compact-valued $\Gamma: X\to P_{cl}[\Omega]$, there is no difference between weak measurability and measurability.

The fundamental result of Kuratowski and Ryll-Nardzewski, [@K-R Theorem 1, p. 398], implies that if $\Gamma: X\to P_{cl}[\Omega]$ is weakly measurable, then $\Gamma$ has a measurable selection, i.e., there exists a single-valued map $\gamma : X\to \Omega$ such that $\gamma(x)\in \Gamma(x)$ for all $x\in X$ and $\gamma$ is $({\cal B}_X, {\cal B}_\Omega)$-measurable: for every Borel set $A\subset \Omega$, $\gamma^{-1}(A)$ is a Borel set in $X$.

We introduce now abstract integral funnels $S(t_0, a)$ that have the properties prompted by the properties of integral funnels of solutions of equation .

$S(t_0, a)$, where $t_0\in [0, +\infty)$ and $a\in X$, is a family of abstract integral funnels on the space $X$ if, for every $t_0\ge 0$, $S(t_0, \cdot) : X\to P_{cl}[\Omega]$ is a set-valued map with the following properties.

1.  For every $a\in X$, $S(t_0, a)$ is a non-empty 