[en] A muon collider represents the ideal machine to reach very high center-of-mass energies and luminosities by colliding elementary particles. This is the result of the low level of beamstrahlung and synchrotron radiation compared to linear or circular electron-positron colliders. In contrast with other lepton machines, the design of a detector for a multi-TeV muon collider requires detailed knowledge of the interaction region due to the significant backgrounds created by muon beam decays in the collider ring. The physics reach can be properly evaluated only when the detector performance in such an environment is determined. In this work, the backgrounds generated by muon beams of $750$ GeV are characterized and the performance of the tracking system and the calorimeter detector is illustrated. Solutions to minimize the effect of the beam-induced backgrounds are discussed and applied to obtain track and jet reconstruction performance. The ${\mathrm{\mu <!--\; ??\; -->}}^{+}{\mathrm{\mu <!--\; ??\; -->}}^{-<!--\; ???\; -->}\to <!--\; ???\; -->H\mathrm{\nu <!--\; ??\; -->}\stackrel{¯<!--\; ??\; -->}{\mathrm{\nu <!--\; ??\; -->}}\to <!--\; ???\; -->b\stackrel{¯<!--\; ??\; -->}{b}\mathrm{\nu <!--\; ??\; -->}\stackrel{¯<!--\; ??\; -->}{\mathrm{\nu <!--\; ??\; -->}}$ process is fully simulated and reconstructed to demonstrate that physics measurements are possible in this harsh environment. The measurement precision for the Higgs boson coupling to $b\stackrel{¯<!--\; ??\; -->}{b}$ is evaluated for $\sqrt{s}=1.5$, 3, and 10 TeV and compared to other proposed machines.