  We generalize the concept of mutually unbiased bases (MUB) to measurements
which are not necessarily described by rank one projectors. As such, these
measurements can be a useful tool to study the long standing problem of the
existence of MUB. We derive their general form, and show that in a finite,
$d$-dimensional Hilbert space, one can construct a complete set of $d+1$
mutually unbiased measurements. Beside of their intrinsic link to MUB, we show,
that these measurements' statistics provide complete information about the
state of the system. Moreover, they capture the physical essence of
unbiasedness, and in particular, they satisfy non-trivial entropic uncertainty
relation similar to $d+1$ MUB.
