The spaces L¹(m) of all m-integrable (resp. $L¹_{w}(m)$ 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, $L¹_{w}(m)$ is always a complex vector lattice; this is not necessarily so for L¹(m). To identify precisely when L¹(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 $L¹(m) ⊆ L¹_{w}(m)$ (which always holds) is proper, then L¹(m) and $L¹_{w}(m)$ contain lattice-isomorphic copies of the complex Banach lattices c₀ and $ℓ^∞$, respectively. On the other hand, whenever L¹(m) contains an isomorphic copy of c₀, merely in the lcHs sense, then necessarily $L¹(m) ⊊ L¹_{w}(m)$. Moreover, the X-valued integration operator Iₘ: f ↦ ∫ fdm, for f ∈ L¹(m), then fixes a copy of c₀. For X a Banach space, the validity of $L¹(m) = L¹_{w}(m)$ turns out to be equivalent to Iₘ being weakly completely continuous. A sufficient condition for this is the (q,1)-concavity of Iₘ for some 1 ≤ q < ∞. This criterion is fulfilled when Iₘ belongs to various classical operator ideals. Unlike for $L¹_{w}(m)$, the space L¹(m) can never contain an isomorphic copy of $ℓ^{∞}$. 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.