u)(t_2) = u(t_2; t_1, u(t_1; t_0, a))$. On the other hand, $(\sigma_{t_1 - t_0} u)(t_2) = u(t_2; t_0, a)$ by the definition of the shift operator $\sigma_\tau$. This establishes the semigroup property of the measurable selection $u(t; t_0, a)$.

Once the (measurable) selection $u$ is found, we define the process $U(t_1, t_0): X\to X$ for $t_1\ge t_0\ge 0$, by the formula $U(t_1, t_0)(a) = u(t_1; t_0, a)$. As we have shown, the map $a\mapsto \{u(\cdot; t_0, a)\}$ from $X$ to $2^\Omega$ is measurable and singleton-valued. By the Kuratowski-Ryll-Nardzewski selection theorem the map $a\mapsto u(\cdot; t_0, a)$ from $X$ to $\Omega^{t_0}$ is $({\cal B}_X,{\cal B}_\Omega)$-measurable. The map $U(t_1, t_0): X\to X$ is the composition of the map $a\mapsto u(\cdot; t_0, a)$ and the evaluation map $\pi_{t_1} : \Omega^{t_0}\owns w\to \pi_{t_1}(w) = w(t_1)\in X$, which is continuous. Consequently, $U(t_1, t_0)$ is Borel measurable. This completes the proof.

Local processes {#loc-proc}
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Let $X$ be a separable complete metric space as in the previous section. The integral funnels $S(t_0, a)$ will now be local. This means that to every initial state $a\in X$ and every initial moment $t_0$ corresponds a strictly positive number $T(t_0, a)$, the *terminal time*. The paths in the funnel 