Let $E$ be a complete locally convex space and let $(X, \Sigma, \mu)$ be a measure space where $\mu$ is a Radon measure. Then the space $L^{1}(X,E)$ is defined as a the completion of the space $S(X,E)$ of simple functions with respect to the topology defined by the seminorms $$ f \mapsto \int_{X} p(f(x))\,\mathrm{d}\mu(x)$$ where $p$ ranges over a family of seminorms defining the topology of $E$.
Under which assumptions (more general than Fréchet spaces) is it known wheter the elements of this completion can be regarded as functions $X\to E$?
Edit: One motivation for this question is that in the second volume of Köthe's "Topological vector spaces" he writes
For arbitrary locally convex $F$ the situation is more complex. There are cases in which not all elements of $L^{1}_{X,\mu}\{F\}$, $F$ complete, are representable as classes of $F$-valued functions.
I am particularly interested in the question of whether these problems can already arise in locally convex Suslin spaces, where most of the problems in the context of vector-valued integration do not appear.
Edit 2: The precise definition of $L_{X,\mu}^{1}\{F\}$ (as used by Köthe) is the following: Denote by $S(F)$ the space of (up to equality almost everywhere) the functions of the form $$ s(t) = \sum_{i=1}^{n} \chi_{A_i}(t) y_i$$ where the $A_i$ are pairwise disjoint measurable sets and such that for every continuous seminorm $q$ the sum $\sum_{i=1}^{m} \mu(A_i) q(y_i)$ is finite. We equip this space with the topology generated by the seminorms $$\pi_q(s) = \int_{X} q(s(t))\,\mathrm{d}\mu = \sum_{i=1}^{m} \mu(A_i) q(y_i).$$ The space $L_{X,\mu}^{1}\{F\}$ is defined as the completion of $S(F)$.