\Omega^\tau \to \Omega\,,\quad (\Game_\tau v)(t) = \begin{cases} 
v(\tau) & \text{when} 0\le t < \tau\,, \\ 
v(t) & \text{when} t \ge \tau\,. 
\end{cases}$$ We will also use the notation $\Omega^\tau_a$ for all the paths in $\Omega^\tau$ staring at the point $a$, i.e., $v(\tau) = a$ if $v\in \Omega^\tau_a$. On occasion, it will be convenient to specify $\tau$ and $a$ in the notation for a path, e.g., $v(t; \tau, a)$.

The integral funnels $S(t_0, a)$ will be subsets of the set $\Omega^{t_0}_a$. Denote by $P_{cl}[\Omega]$ the space of all (bounded) closed subsets of $\Omega$ endowed with the Vietoris topology (= exponential topology), see [@Kuratowski], [@B], and [@A-B] for details on set-valued maps, their properties and, in particular, measurability. In this presentation, all set-valued maps from $X$ to $\Omega$ will have non-empty closed values and will be viewed as maps from the measurable space $(X, {\cal B}_X)$ to $P_{cl}[\Omega]$ with the Vietoris topology. If $\Gamma$ is such a map and $A\subset \Omega$, define $$\Gamma^{-}(A) = \{x\in X : \Gamma(x)\cap A \neq\emptyset\}\,.$$ For historical reasons, there are several confusingly similar notions of measurability of set-valued maps. A map $\Gamma: X\to P_{cl}[\Omega]$ is *weakly measurable* if $\Gamma^{-}(G)\in {\cal B}_X$ for every open set $G\subset\Omega$. $\Gamma$ 