We study functional and geometric inequalities on complete Finsler measure spaces with the weighted Ricci curvature \(\textrm{Ric}_\infty \) bounded below. We first obtain some local uniform Poincaré inequalities and Sobolev inequalities. Then, we prove a mean value inequality for nonnegative subsolutions of elliptic equations. Further, we derive local and global Harnack inequalities for positive harmonic functions. Finally, we establish a global gradient estimate for positive harmonic functions on forward complete non-compact Finsler measure spaces. Besides, as an application of the mean value inequality, we prove a Liouville type theorem for harmonic functions.
