Beyond Lebesgue and Baire IV: Density topologies and a converse Steinhaus-Weil Theorem

The theme here is category-measure duality, in the context of a topological group. One can often handle the (Baire) category case and the (Lebesgue, or Haar) measure cases together, by working bi-topologically: switching between the original topology and a suitable refinement (a density topology). This prompts a systematic study of such density topologies, and the corresponding $\sigma$-ideals of negligibles. Such ideas go back to Weil's classic book, and to Hashimoto's ideal topologies. We make use of group norms, which cast light on the interplay between the group and measure structures. The Steinhaus-Weil interior-points theorem (`on $AA^{-1}$') plays a crucial role here; so too does its converse, the Simmons-Mospan theorem.