Measurability of automorphisms of topological groups

Relations between the measurability and continuity of algebraic automorphisms of topological groups depending on the types of groups are examined. Various cases are considered and theorems on the continuity of measurable automorphisms are proved; for instance, such theorems are proved for separable locally compact groups and automorphisms measurable with respect to nonnegative Haar measures. On the other hand, examples of nonmetrizable nonseparable compact groups with Haar measures and of non-locally-compact separable metrizable groups with measures μ quasi-invariant with respect to dense subgroups admittings μ-measurable discontinous automorphisms are given. Translated fromMatenmaticheskie Zametki, Vol. 68, No. 1, pp. 105–112, July, 2000. An erratum to this article can be found online at .