Having had no (proper) answer to *this question*, I formulate the remaining case as a new question as follows. With $I=[0,1]$, let $E$ be a separable (real) Banach space, and let $\gamma:I\to E$ be continuous. *Do there* then *exist* sequences $\boldsymbol c,\boldsymbol t\in I^{\ \mathbb N_0}$, $\boldsymbol c(i)=c_i$ and $\boldsymbol t(i)=t_i$, with

(1) $\quad\mathbb R\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^kc_i=1 \quad$ and

(2) $\quad E\ \text{-}\ \lim_{\ k\to\infty\ }\big(k^{-1}\sum_{i=0}^{k-1}\gamma(k^{-1}i)\big) = E\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^k(c_i\gamma(t_i)) \quad$ ?

Either a (sketch of a) proof of the positive case or a counterexample is welcome. Countable or σ−convexity has also been considered in *this question*.

here it is in MR0808401:$C$ is measure convex if each inner regular probability measure on C has a barycenter and that barycenter belongs to $C$. ....here it is in MR1009196:A subset of a Banach space is said to be ... measure-convex if it contains the closed convex hull of each of its compact subsets, – Gerald Edgar May 5 '11 at 13:22