  The Muon Collider is one of the most promising future collider facilities
with the potential to reach multi-TeV center-of-mass energy and high
luminosity. Due to the significant Higgs boson production cross section in muon
collisions at these high energies, the collider can be considered a Higgs
factory. It holds the capability to significantly advance our understanding of
the Higgs sector to an unprecedented level of precision. However, the presence
of beam-induced background resulting from the decay of the beam muons poses
unique challenges for detector development and event reconstruction. In this
paper, the prospects for various measurements of the Higgs boson production
cross sections at a $\sqrt{s}=3$ TeV collider are presented using a detailed
detector simulation in a realistic environment. The study demonstrates the
feasibility of achieving high precision measurements of the Higgs boson
production cross sections with the current state-of-the-art detector design. In
addition, the paper discusses the detector requirements necessary for obtaining
such resolutions and for measuring the Higgs trilinear self-coupling.
