  We calculate in closed analytical form the one-photon loop radiative
corrections to muon Compton scattering $\mu^- \gamma \to \mu^- \gamma $.
Ultraviolet and infrared divergencies are both treated in dimensional
regularization. Infrared finiteness of the (virtual) radiative corrections is
achieved (in the standard way) by including soft photon radiation below an
energy cut-off $\lambda$. We find that the anomalous magnetic moment
$\alpha/2\pi$ provides only a very small portion of the full radiative
corrections. Furthermore, we extend our calculation of radiative corrections to
the muon-nucleus bremsstrahlung process (or virtual muon Compton scattering
$\mu^-\gamma_0^* \to \mu^- \gamma $). These results are particularly relevant
for analyzing the COMPASS experiment at CERN in which muon-nucleus
bremsstrahlung serves to calibrate the Primakoff scattering of high-energy
pions off a heavy nucleus with the aim of measuring the pion electric and
magnetic polarizabilities. We find agreement with an earlier calculation of
these radiative corrections based on a different method.
