Abstract Measure Theory, Integration on Metric Measure Spaces, and Applications to Data Analysis in Sub-Saharan African Health Systems

This dissertation develops original contributions to the theory of integration on metric measure spaces in the setting of Cheeger-type differential structures and Hajlasz-Sobolev spaces, establishing new results on the relationship between the Poincare inequality, doubling conditions, and the existence of first-order differential calculi on abstract metric spaces, with a long-form application component addressing the analysis of high-dimensional health outcomes data from sub-Saharan African health information systems including Nigeria's Health Management Information System. Metric measure space theory provides a coordinate-free generalisation of classical analysis that accommodates the fractal, hierarchical, and non-Euclidean structures arising in complex data settings, and its application to health data analysis addresses the lack of a sufficiently general mathematical foundation for clustering, dimension reduction, and regression on health system data with complex relational structures. The dissertation proves new characterisations of the upper gradient Sobolev space W1,p on locally compact doubling metric measure spaces satisfying a (1,p)-Poincare inequality, and establishes the equivalence of the Newton-Sobolev and Hajlasz-Sobolev space definitions under minimal regularity hypotheses. A metric-space-valued extension of the Cheeger-Gromov compactness theorem for spaces with Ricci curvature bounded below is developed and applied to establish convergence of data-driven graph-based Laplacians to continuous spectral operators. Applications to clustering of local government area-level health indicator profiles from the Nigeria Demographic and Health Survey are developed. Keywords: metric measure spaces, Sobolev spaces, Poincare inequality, health data analysis, Nigeria