  We present spin-resolved measurements of the local density of states in Si
doped GaAs. Both spin components exhibit strong mesoscopic fluctuations. In the
magnetic quantum limit, the main features of the spin-up and spin-down
components of the local density of states are found to be identical apart from
Zeeman splitting. Based on this observation, we introduce a mesoscopic method
to measure the $g$-factor in a material where macroscopic methods are severely
restricted by disorder. Differences between the spin-up and spin-down
components are discussed in terms of spin relaxation due to spin-orbit
coupling.
