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Abstract

The spaces $L^1(m)$ of all $m$-integrable (resp. $L^1_w(m)$ of all scalarly $m$-integrable) functions for a vector measure $m$, taking values in a complex locally convex Hausdorff space $X$ (briefly, lcHs), are themselves lcHs for the mean convergence topology. Additionally, $L^1_w(m)$ is always a complex vector lattice; this is not necessarily so for $L^1(m)$. To identify precisely when $L^1(m)$ is also a complex vector lattice is one of our central aims. Whenever $X$ is sequentially complete, then this is the case. If, additionally, the inclusion $L^1(m) \subseteq L^1_w(m)$ (which always holds) is proper, then $L^1(m)$ and $L^1_w(m)$ contain lattice-isomorphic copies of the complex Banach lattices $c_0$ and $\ell^\infty$, respectively. On the other hand, whenever $L^1(m)$ contains an isomorphic copy of $c_0$, merely in the lcHs sense, then necessarily $L^1(m) \subsetneq L^1_w(m)$. Moreover, the $X$-valued integration operator $ I_m : f \mapsto \int f \, d m $, for $f \in L^1 (m)$, then fixes a copy of $c_0$. For $X $ a Banach space, the validity of $L^1(m) = L^1_w (m)$ turns out to be equivalent to $I_m$ being weakly completely continuous. A sufficient condition for this is the $(q,1)$-concavity of $I_m$ for some $1 \le q < \infty $. This criterion is fulfilled when $I_m$ belongs to various classical operator ideals. Unlike for $L^1_w(m)$, the space $L^1(m)$ can never contain an isomorphic copy of $\ell^\infty $. A rich supply of examples and counterexamples is presented. The methods involved are a hybrid of vector measure/integration theory, functional analysis, operator theory and complex vector lattices.