Having had no (proper) answer to [*this question*][1], 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

<blockquote>
(1) $\quad\mathbb R\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^kc_i=1 \quad$ and</blockquote><blockquote>(2) $\quad
E\ \text{-}\ \lim_{\ k\to\infty\ }\big(k^{-1}\sum_{i=0}^{k-1}\gamma(k^{-1}i)\big) =
E\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^k(c_i\gamma(t_i)) \quad$ ?</blockquote>

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*][2].


  [1]: http://mathoverflow.net/questions/57813/stability-of-convex-sets-w-r-t-integration-over-0-1
  [2]: http://mathoverflow.net/questions/56161/infinite-convex-combinations-in-a-banach-space