Automorphism groups for measurable spaces

In [1], the notion of a rigid separable measurable space was defined, and such spaces were shown to exist. We expand upon this idea and ask what are the possible automorphism groups for such spaces. We show that there is only one such non-trivial group which is Abelian (a countable product of two element groups), and that this group is realised as the automorphism group of some separable space. A particular class of such spaces is characterised in terms of rigid components.Finally, an example of a measurable space which is rigid in the strict sense is constructed. This answers a question of K.P.S. Bhaskara Rao and B.V. Rao.