is measurable. This is a new type of selection theorems (for measurable selection results see, e.g., [@A-B Section 18.3], [@H-P], and the surveys [@Wa; @Io]). Our results were motivated by the Markov selection theorems, see [@Krylov; @S-V; @F-R; @G-R-Z].

Here, we extend the semiflow selection theorem of [@C; @C-K] to non-autonomous equations and processes. If there is no uniqueness for , but the solutions to the initial-value problem exist forward in time, we have the integral funnels $S(t_0, a)$ formed by all the solution $u(t; t_0, a)$, $t\ge t_0$, such that $u(t_0; t_0, a) = a$. The question we ask is whether there exists a measurable selection $(t_0, a) \mapsto u(\cdot; t_0, a)\in S(t_0, a)$ such that $u(t; t_0, a)$ satisfies . The answer again is yes under the right assumptions. After our previous work [@C; @C-K], this answer is not surprising if we are concerned with solutions of the non-autonomous equation : one could replace with the equivalent autonomous system \[n-aut-2\] = g(t, u), = 1. Then, if we apply our results from [@C; @C-K] and obtain a semigroup ${\tilde U}(s)$, $s\ge 0$, on $X\times [0, +\infty)$ corresponding to , the maps $U(t_1, t_0)(a) = {\tilde U}(t_1)(a, t_0)$ would 