  Determining the measurement uncertainty region is a difficult problem for
generic sets of observables. For this reason the literature on exact
measurement uncertainty regions is focused on symmetric sets of observables,
where the symmetries are used to simplify the calculation. We provide a
framework to systematically exploit available symmetries, formulated in terms
of covariance, to simplify problems of measurement uncertainty. Our key result
is that for a wide range figures of merit the optimal compatible approximations
of covariant target observables are themselves covariant. This substantially
simplifies the problem of determining measurement uncertainty regions for cases
where it applies, since the space of covariant observables is typically much
smaller than that of all observables. An intermediate result, which may be
applicable more broadly, is the existence and characterisation of a
covariantisation map, mapping observables to covariant observables. Our
formulation is applicable to finite outcome observables on separable Hilbert
spaces. We conjecture that the restriction of finite outcomes may be lifted,
and explore some of the features a generalisation must have. We demonstrate the
theorem by deriving measurement uncertainty regions for three mutually
orthogonal Pauli observables, and for phase space observables in arbitrary
finite dimensions.
