Souslin measurable homomorphisms of topological groups

1. In a recent paper [18], A. Kleppner proved that every Haar measurable homomorphism 4 : G ~ H between locally compact groups G, H is continuous. This result also follows from a measure theoretic result by D. H. Fremlin [8], (see also [9, 19]), which provides a technique to prove measurability of uncountable unions of measurable sets. The purpose of this note is to present a different and new approach to continuity of measurable homomorphisms, based on Souslin and Borel rather than Haar measurability. This slight investment on part of the measurability of the homomorphism 4~ allows for weakening considerably the conditions to be imposed on the groups G, H. In particular, the presence of the Haar measure, which is essential in the approach of [18] resp. [8, 9], is no longer needed to establish our continuity result. Our technique is completely different from the methods used in [18] (resp. [8, 9]). It gives rise also to a generalization of Banach's classical result [2] stating that Borel measurable homomorphisms between Polish groups G, H are continuous. We prove that every Souslin measurable homomorphism q5 : G ~ H from a paracompact Cech complete group G into any topological group H is continuous (Theorem 1). Some consequences of this result such as a theorem on joint continuity of separately measurable homomorphisms and a version of the Sousiin graph theorem are included. There is, of course, a variety of results dealing with continuity of homomorphisms of topological groups measurable in one sense or other, in contrast with our present approach, these all need countability type assumptions on at least one of the groups G, H. We just mention the following list [2, 5, 10, 13, 15, 17, 18] of references on Borel and Souslin graph theorems and continuity theorems for measurable homomorphisms. Further references are [1, 3, 5, 12, 16, 22, 23, 25, 26], where closed graph theorems are treated. An interesting paper is [27], where the continuity of sequentially continuous homomorphisms between locally compact groups is discussed.