Integral in a σ−convex set. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T15:14:55Z http://mathoverflow.net/feeds/question/63922 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/63922/integral-in-a-convex-set Integral in a σ−convex set. TaQ 2011-05-04T15:22:42Z 2011-05-04T19:34:03Z <p>Having had no (proper) answer to <a href="http://mathoverflow.net/questions/57813/stability-of-convex-sets-w-r-t-integration-over-0-1" rel="nofollow"><em>this question</em></a>, 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. <em>Do there</em> then <em>exist</em> sequences $\boldsymbol c,\boldsymbol t\in I^{\ \mathbb N_0}$, $\boldsymbol c(i)=c_i$ and $\boldsymbol t(i)=t_i$, with</p> <p><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></p> <p>Either a (sketch of a) proof of the positive case or a counterexample is welcome. Countable or σ−convexity has also been considered in <a href="http://mathoverflow.net/questions/56161/infinite-convex-combinations-in-a-banach-space" rel="nofollow"><em>this question</em></a>.</p> http://mathoverflow.net/questions/63922/integral-in-a-convex-set/63944#63944 Answer by Gerald Edgar for Integral in a σ−convex set. Gerald Edgar 2011-05-04T19:34:03Z 2011-05-04T19:34:03Z <p>Counterexample. $E = L^2[0,1]$ and $\gamma \colon [0,1] \to E$ defined by $\gamma(x) = 1_{[0,x]}$, the characteristic function of interval $[0,x]$. Then $\gamma$ is continuous, in fact $\|\gamma(x) - \gamma(y)\| = \sqrt{|y-x|}$. Now suppose $c_i$ and $t_i$ are as given. Let $$u := E\ \text{-}\ \lim_{\ k\to\infty\ }\big(k^{-1}\sum_{i=0}^{k-1}\gamma(k^{-1}i)\big)$$ and $$v := E\ \text{-}\ \lim_{\ k\to\infty\ }\sum_{i=0}^k(c_i\gamma(t_i))$$ Then $u(t) = 1-t$ for $t \in [0,1]$. Note $u$ is continous. But (assuming the $t_i$ are distinct) $v$ has a jump of size $c_i$ at $t_i$ for all $i$, so $v$ is certainly not continuous. Not even equal a.e. to a continuous function.</p>