### Spectral measures and automatic continuity.

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CONTENTSIntroduction...............................................................................51. Preliminaries.........................................................................72. Relative weak compactness of the range............................133. Closed spectral measures...................................................164. Spectral measures and B.a.'s of projections........................22References..............................................................................45...

Associated with every vector measure m taking its values in a Fréchet space X is the space L(m) of all m-integrable functions. It turns out that L(m) is always a Fréchet lattice. We show that possession of the AL-property for the lattice L(m) has some remarkable consequences for both the underlying Fréchet space X and the integration operator f → ∫ f dm.

The spaces L¹(m) of all m-integrable (resp. $L{\xb9}_{w}\left(m\right)$ 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, $L{\xb9}_{w}\left(m\right)$ 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,...

A characterization is given of those Banach-space-valued vector measures m with finite variation whose associated integration operator Iₘ: f ↦ ∫fdm is compact as a linear map from L¹(m) into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures m (with finite variation) such that Iₘ is compact, and other m (still with finite variation) such that Iₘ is not compact. If m has infinite variation, then Iₘ is never compact.

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