The Spin Muon Collaboration (SMC) measures the spin dependent structure function g\sb1 of the proton and nentron by measuring the scattering asymmetry of polarized 200 GeV muons off polarized protons and deuterons. The structure functions enable tests of theoretical sum rules, and a measurement of the spin contribution of the quarks to the nucleon. The uncertainty of the muon beam polarization was a major source of error in preliminary measurements of proton structure functions. A muon polarimeter measuring the shape of the Michel spectrum of positrons from muon decay was built. In this polarimeter muons enter and are allowed to decay (μ\sp+→e\sp+ν\sbeν¯\sbμ) in a 35 meter length. The shape of the momentum spectrum of electrons is sensitive to the muon polarization. The decay positrons are momentum-analyzed and the measured spectrum is fit to the Michel formula to determine the polarization. A data sample with a μ\sp− beam was used to estimate the effects of background events in the spectrum. Careful analysis of the polarimeter data determined the polarization to within 3%. The muon polarization was found to be stable in time and to vary with muon momentum. This variation will be included in the structure function analysis. A second polarimeter measuring the scattering asymmetry of polarized muons off polarized electrons obtained consistent results. The two independent polarization measurements were combined to give a polarization of $-$0.778 ± 0.019 at 186.9 GeV. With the improved structure function measurements, the Bjorken sum rule was tested and confirmed. Assuming that the gluons are unpolarized, the contribution of the quarks to the nucleon spin was estimated to be 20%, and the strange quark sea negatively polarized.