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Autocontinuity and convergence theorems for the Choquet integral
Our aim is to provide some convergence theorems for the Choquet integral with respect to various notions of convergence. For instance, the dominated convergence theorem for almost uniform convergence is related to autocontinuous set functions. Autocontinuity can also be related to convergence in measure, strict convergence or mean convergence. Whereas the monotone convergence theorem for almost uniform convergence is related to monotone autocontinuity, a weaker version than autocontinuity.
Geometric properties of noncommutative symmetric spaces of measurable operators and unitary matrix ideals
This is a survey article of geometric properties of noncommutative symmetric
spaces of measurable operators , where is a
semifinite von Neumann algebra with a faithful, normal, semifinite trace
, and is a symmetric function space. If is a symmetric
sequence space then the analogous properties in the unitary matrix ideals
are also presented. In the preliminaries we provide basic definitions and
concepts illustrated by some examples and occasional proofs. In particular we
list and discuss the properties of general singular value function,
submajorization in the sense of Hardy, Littlewood and P\'olya, K\"othe duality,
the spaces , , the identification between
and for some symmetric function space , the
commutative case when is identified with for
isometric to with the standard integral trace, trace
preserving -isomorphisms between and a -subalgebra of
, and a general method of removing the assumption of
non-atomicity of . The main results on geometric properties are
given in separate sections. We present the results on (complex) extreme points,
(complex) strict convexity, strong extreme points and midpoint local uniform
convexity, -extreme points and -convexity, (complex or local) uniform
convexity, smoothness and strong smoothness, (strongly) exposed points,
(uniform) Kadec-Klee properties, Banach-Saks properties, Radon-Nikod\'ym
property and stability in the sense of Krivine-Maurey. We also state some open
problems
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