Convergence in measure with respect to a matrix-valued measure and some matrix completion problems

Abstract

For a matrix-valued measure M we introduce a notion of convergence in measure M, which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of M. Let S be a subset of the set of matrices of given size. It is easy to see that the set of S -valued measurable functions is closed under convergence in measure with respect to a matrix-valued measure if and only if S is a ρ -closed set, i.e. if and only if S P is closed for any orthoprojector P. We discuss the behaviour of ρ -closed sets under operations of linear algebra and the ρ -closedness of particular classes of matrices.