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 $\mathcal S$ be a subset of the set of matrices of given size.
It is easy to see that the set of $\mathcal S$-valued measurable functions
is closed under convergence in measure with respect to a matrix-valued measure
if and only if $\mathcal S$ is a $\rho$-closed set, i.e. if and only if
$\mathcal SP$ is closed for any orthoprojector $P$. We discuss the behaviour
of $\rho$-closed sets under operations of linear algebra and the $\rho$-closedness
of particular classes of matrices.