Abstract: In this letter, we evaluate the 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 (t) over bar. This matchedcalculation includes the QCD enhancement of the cross section around the t (t) over bar 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 root s = 380 GeV, and with a precision of approximately 150 MeV at the ILC, with L = 4 ab(-1) at root s = 500GeV. 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)(MSR)(R) for scales R < m(t).