Klotz, Lutz; Wang, Dong Metric properties of convergence in measure with respect to a matrix-valued measure. (English) Zbl 1324.28002 Acta Math. Acad. Paedagog. Nyházi. (N.S.) 30, 67-78 (2014). 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 Keywords:convergence in measure; matrix-valued measure; metric space; dense set; shift invariant subspace PDFBibTeX XMLCite \textit{L. Klotz} and \textit{D. Wang}, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 30, 67--78 (2014; Zbl 1324.28002) Full Text: EMIS