Dissertation is titled "Weakly Montone Functions on Metric Measure Spaces". We introduce the theory of weakly monotone functions on metric measure spaces $(X,d,\mu)$. We develop analytic properties of these functions on metric measure spaces that are complete and support a $p-$Poincar\'e inequality; these properties agree with weakly monotone functions in Euclidean space. We prove that weakly monotone functions are locally essentially bounded, admit representatives that are continuous outside a set of small Hausdorff measure, and are differentiable in the sense of Cheeger on the metric space almost everywhere. In the limiting case, we prove that these functions admit a continuous representative.

Further, we prove that each Newtonian function $u\in N^{1,p}(x)$ can be approximated by locally weakly monotone functions via truncation. This argument relies mainly on the lattice properties of the Newtonian space and the lattice supremum. The minimal upper gradients of the approximations $u_j$ are zero almost everywhere in the set where $\{x\in X: u_j(x)\neq u(x)\}$. Utilizing this property, we exhibit a class of variational problems whose solutions (when they exist and are unique) must give rise to a weakly monotone function. This class includes the p-Dirichlet integrals. We also prove that integrands of this type respect the convergence of the approximation by weakly monotone functions. Both integrands depending on the upper gradients and the Cheeger derivatives are considered.

Committee Chair and Advisor: Dr. Piotr Hajlasz