Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the membership of Iₘ in various classical operator ideals(e.g., the compact, p-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space X may also play a crucial role. Of particular importance in this regard is whether or not X contains an isomorphic copy of the classical sequence space ℓ¹. The compact range property of X is also relevant.