  The spectrum of electrons from muons decaying in an atomic bound state is
significantly modified by their interaction with the nucleus. Somewhat
unexpectedly, its first measurement, at the Canadian laboratory TRIUMF,
differed from basic theory. We show, using a combination of techniques
developed in atomic, nuclear, and high-energy physics, that radiative
corrections eliminate the discrepancy. In addition to solving that outstanding
problem, our more precise predictions are potentially useful for interpreting
future high-statistics muon experiments that aim to search for exotic
interactions at $10^{-16}$ sensitivity.
