A ‘2-group’ is a category equipped with a multiplication satisfying laws like those of a
group. Just as groups have representations on vector spaces, 2-groups have representations
on ‘2-vector spaces’, which are categories analogous to vector spaces. Unfortunately, Lie 2-
groups typically have few representations on the finite-dimensional 2-vector spaces introduced
by Kapranov and Voevodsky. For this reason, Crane, Sheppeard and Yetter introduced certain
infinite-dimensional 2-vector spaces called ‘measurable categories’ (since they are closely related
to measurable fields of Hilbert spaces), and used these to study infinite-dimensional represen￾tations of certain Lie 2-groups. Here we continue this work. We begin with a detailed study
of measurable categories. Then we give a geometrical description of the measurable represen￾tations, intertwiners and 2-intertwiners for any skeletal measurable 2-group. We study tensor
products and direct sums for representations, and various concepts of subrepresentation. We
describe direct sums of intertwiners, and sub-intertwiners—features not seen in ordinary group
representation theory. We study irreducible and indecomposable representations and intertwin￾ers. We also study ‘irretractable’ representations—another feature not seen in ordinary group
representation theory. Finally, we argue that measurable categories equipped with some extra
structure deserve to be considered ‘separable 2-Hilbert spaces’, and compare this idea to a ten￾tative definition of 2-Hilbert spaces as representation categories of commutative von Neumann
algebras.