Riemannian manifolds with measure, and more general metric spaces with measure have been studied under different guises for decades. In this talk I'll present a new geometric approach to these spaces that introduces a torsion free affine connection, as opposed to a measure, as the fundamental object of study. This approach allows us to connect the notion of a manifold with measure and the Bakry-Emery notion of manifold with density. This allows us to extend the work on comparison results to the new case of negative synthetic dimension, as well as reveals a number of new structures for future study. This is joint work with Will Wylie.