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 \[zeta-2\] (w) = \_0\^[T(t\_0, a)]{} 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) and define \[V\] V\_= {vS(t\_0, a): (v) = m\_(t\_0, a)}. The way we treat measurability by re-parametrizing the paths (see property [**LS2**]{}), allows us to apply the measurable maximum theorem, [@A-B Theorem 18.19], and conclude that $V_\zeta[S(t_0, a)]$ 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. This shows that the family of sets $V_\zeta[S(t_0, a)]$ has properties [**LS1**]{} and [**LS2**]{}. Suppose $u(\cdot; t_0, a)\in V_\zeta[S(t_0, a)]$ and consider the shifted path $\sigma_\tau u$. By property [**LS3**]{}, $\sigma_\tau u\in S(t_0+\tau, u(t_0 + \tau; t_0, a))$. Let us show that $\sigma_\tau u$ maximizes $\zeta$ in the set $S(t_0+\tau, u(t_0 + \tau; t_0, a))$. Pick any path $v$ in $S(t_0+\tau, u(t_0 + \tau; t_0, a))$ and consider the spliced path $w = u \underset{t_0+\tau}{\bowtie} v$, which, 