  An expression for the Dalitz plot of the semileptonic decay of a charged
kaon, including radiative corrections to order O[\alpha \pi)(q/M_1)], where q
is the four-momentum transfer and M_1 is the mass of the decaying kaon, is
obtained. Contributions of both the three- and four-body regions are accounted
for. Besides, the emitted muon is considered to be polarized so the analysis is
also focused on evaluating the radiative corrections to the longitudinal,
transverse, and normal polarization components of the muon. The final formulas,
with the triple integration of the bremsstrahlung photon variables ready to be
performed numerically, are general enough to be used in model-independent
experimental analyses whether or not the real photon is discriminated. With the
numerical values of the weak form factors and slope parameters of the process,
the radiative corrections to the components of the muon polarization are found
to be very small compared to their respective uncorrected values.
