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Metric properties of convergence in measure with respect to a matrix-valued measure. (English) Zbl 1324.28002

Summary: A notion of convergence in measure with respect to a matrix-valued measure \(M\) is discussed and a corresponding metric space denoted by \(L_0(M)\) is introduced. There are given some conditions on \(M\) under which \(L_0(M)\) is locally convex or normable. Some density results are obtained and applied to the description of shift invariant sub-modules of \(L_0(M)\) if \(M\) is defined on the \(\sigma\)-algebra of Borel sets of \((-\pi; \pi]\).

MSC:

28A20 Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46A16 Not locally convex spaces (metrizable topological linear spaces, locally bounded spaces, quasi-Banach spaces, etc.)
42A10 Trigonometric approximation
47A15 Invariant subspaces of linear operators
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