The discovery of the Higgs particle at 125 GeV is demanding a detailed
knowledge of the properties of this fundamental component of the Standard
Model. Various proposals of electron and muon colliders have been put
forward for precision studies of the partial widths of the various decay
channels. It will be shown that in the case of a Higgs factory through a
muon collider, sizeable radiative effects - of order of 50% - must be
carefully taken into account for a precise measurement of the leptonic and
total widths of the Higgs particle. Similar effects do not apply in the
case of Higgs production in electron-positron colliders.