Automatic continuity of measurable homomorphisms on topological groups

We shall discuss three recent results on automatic continuity of homomorphisms:

1) Every Haar-measurable homomorphism from a locally compact group to an arbitrary topological group is continuous.

2) Every homomorphism with the Baire property from an $\omega$-narrow Cech-complete topological group to an arbitrary topological group is continuous

3) Every Borel homomorphism from a Cech-complete topological group to an arbitrary topological group is continuous. 

The continuity of Haar-measurable homomorphisms on locally compact groups was proved by Kuznetsova in 2012 under Martin's Axiom. Our proofs are based on a new result about analytically nonmeasurable unions of sets in $\sigma$-ideals on Polish spaces, 

which was obtained jointly with Ralowski and Zeberski.