An elementary approach to Haar integration and Pontryagin duality in locally compact abelian groups

We offer an elementary proof of Pontryagin duality theorem for compact and discrete abelian groups. To this end we make use of an elementary proof of Peter-Weyl theorem due to Prodanov that makes no recourse to Haar integral. As a long series of applications of this approach we obtain proofs of Bohr von Neumann’s theorem on almost periodic functions, Comfort-Ross’ theorem on the description of the precompact topologies on abelian groups, and, last but not least, the existence of Haar integral in LCA groups.