I have come across this fact in some paper but I don't remember where exactly: it says that continuous functions defined on a closed interval $[a, b]$ in the set $\Bbb{R}$ of real numbers with values in a non-locally convex topological vector space may fail to be integrable with respect to the Lebesgue measure. Then my question here is:
Question: Are there some published examples of such functions? Or, are there examples of continuous functions defined on $[a, b]$ that are not integrable?
Consider the space $\ell_p$ for $p = 1/4$. This space equipped with distance $\rho(x, y) = \|x - y\|^p$ is not locally convex complete metrizable topological vector space. Denote $e_n$ the vectors of the canonical basis of $\ell_p$, that is $e_1 = (1, 0, 0, ...)$, $e_2 = (0, 1, 0, 0, ...)$, etc. Let $\Delta_n = [2/(2n+1), 1/n]$. The intervals $\Delta_n$ are mutually disjoint and tend to 0. Now define the required function $f: [0, 1] \to \ell_p$ as follows: on every $\Delta_n$ the function $f$ takes the constant value $e_n/n$; between two neighboring intervals $\Delta_n$, $\Delta_{n+1}$ define $f$ by means of linear interpolation (in order to make it continuous), and put $f(0) = 0$.