It is well-known that, if $(\Omega,\mathcal{F},P)$ is an atomless probability space, then $L^1$ is barreled, in the sense that every subset $K$ which is closed, convex, absorbent and balanced is a neighborhood of $0$ ($K$ is called a barrel).
So, what about non closed subsets? Can we find a non closed subset which is convex, absorbent and balanced, but not neighborhood of $0$?
Kind regards
HINT:
Let $E$ be an infinite dimensional space with a norm $p$. Then there exists a norm $q$ on $E$ such that $q \not \preceq p$. ( as opposed to : on a finite dimenisonal space any two norms are equivalent).
Indeed, consider $(e_i)_{i\in I}$ a basis. Consider $f\colon I \to (0, \infty)$ an unbounded function ( you could take $e_n$ a countable subset of $e_i$ and define $f(n) = n$). Define
$$q( \sum_i a_i e_i) = \sum f(i) |a_i| p(e_i)$$
Check that $q$ is a finite norm and $q \not \preceq p$ ( for the last one, enough to check on the $e_i$'s).
Now, as the subset take $$K \colon = \{ v \ | \ q(v) \le 1\}$$