  In this paper, we study functional and geometric inequalities on complete
Finsler measure spaces under the condition that the weighted Ricci curvature
${\rm Ric}_\infty$ has a lower bound. We first obtain some local uniform
Poincar\'{e} inequalities and Sobolev inequalities. Further, we give a mean
value inequality for nonnegative subsolutions of elliptic equations. Finally,
we obtain local and global Harnack inequalities, and then, establish a global
gradient estimate for positive harmonic functions on forward complete
non-compact Finsler measure spaces. Besides, as a by-product of the mean value
inequality, we prove a Liouville type theorem.
