based on an idea from optimization theory that was used by N. V. Krylov in his proof of the Markov selection theorem, [@Krylov].

Let $\vp: X\to [0, 1]$ be a continuous function and let $\lambda$ be a positive real number. For $t_0 \ge 0$ and $a\in X$, define the functional $\zeta$ on $S(t_0, a)$ via the formula \[zeta\] (w) = \_0\^e\^[-t]{} (w(t\_0 + t))dt. This is a continuous functional and it attains its maximum on the compact set $S(t_0, a)$. Denote this maximum by $m_\zeta(t_0, a)$, \[m\] m\_(t\_0, a) = \_[wS(t\_0, a)]{} (w) Define \[V\] V\_= {vS(t\_0, a): (v) = m\_(t\_0, a)}. By the so-called measurable maximum theorem, [@A-B Theorem 18.19], $V_\zeta[S(t_0, a)]$ is a non-empty compact subset of $S(t_0, a)$, and the set-valued map $a \mapsto V_\zeta[S(t_0, a)]$ is measurable. Thus, the family of sets $V_\zeta[S(t_0, a)]$ has properties [**S1**]{} and [**S2**]{} of the abstract funnels. Let us check that it has the remaining two properties [**S3**]{} and [**S4**]{}. Suppose $u\in V_\zeta[S(t_0, a)]$ and consider the shifted path $\sigma_\tau u$. By property [**S3**]{} of the family $S$, $\sigma_\tau u\in S(t_0+\tau, u(t_0 + \tau))$. We have to show that $\sigma_\tau u$ maximizes $\zeta$ in the set $S(t_0+\tau, u(t_0 + \tau))$. 