  Mutually unbiased measurements are a generalization of mutually unbiased
bases in which the measurement operators need not to be rank one projectors. In
a $d$-dimension space, the purity of measurement elements ranges from $1/d$ for
the measurement operators corresponding to maximally mixed states to $1$ for
the rank one projectors. In this contribution, we provide a class of MUM that
encompasses the full range of purity. Similar to the MUB in which the operators
corresponding to different outcomes of the same measurement commute mutually,
our class of MUM possesses this sense of compatibility within each measurement.
This makes the provided class more similar to the MUB, so that the main
difference between them and MUB is due to the purity of the measurement
operators. The spectra of these MUMs provides a way to construct a class of
$d$-dimensional orthogonal matrices which leave the vector of equal components
invariant. Based on this property, and by using the MUM-based entanglement
witnesses, we investigate the role of purity to detect entanglement of
bipartite states.
