  In this letter, we evaluate potential of linear $e^+e^-$ colliders to measure
the top quark mass in radiative events and in a suitable short-distance scheme.
We present a calculation of the differential cross section for production of a
top quark pair in association with an energetic photon from initial state
radiation, as a function of the invariant mass of the $t\bar{t}$ system. This
{\it matched} calculation includes the QCD enhancement of the cross section
around the $t\bar{t}$ production threshold and remains valid in the continuum
well above the threshold. The uncertainty in the top mass determination is
evaluated in realistic operating scenarios for the Compact Linear Collider
(CLIC) and the International Linear Collider (ILC), including the statistical
uncertainty and the theoretical and experimental systematic uncertainties. With
this method, the top quark mass can be determined with a precision of $110$ MeV
in the initial stage of CLIC, with $1$ ab$^{-1}$ at $\sqrt{s} =$ 380 GeV, and
with a precision of approximately $150$ MeV at the ILC, with $L = 4$ ab$^{-1}$
at $\sqrt{s}= 500$ GeV. Radiative events allow measurements of the top quark
mass at different renormalization scales, and we demonstrate that such a
measurement can yield a statistically significant test of the evolution of the
MSR mass $m_t^{\rm MSR}(R)$ for scales $R< m_t$.
