I will speak about a new(ish) generalisation (with Macpherson, Steinhorn, and Wolf) of (MS-)measurable structures via two key examples. The first is the two-sorted structure of an infinite-dimensional vector space over a pseudofinite field, together with a non-degenerate bilinear form, either symmetric or alternating.
These structures were studied by Granger, and shown to have the Tree Property (in fact the Tree Property of the Second Kind). Secondly I will show how to view `free homogeneous structures' as generalised measurable.
By a `free homogeneous structure' I mean a Fraisse amalgam of a class with free amalgamation. For example, the generic triangle-free graph is free homogeneous, and this has the Tree Properties of the first and