Strongly minimal pseudofinite structures

In this note we point out that any strongly minimal pseudofinite structure (or set) is unimodular in the sense of [1], [5], [2], and hence measurable in the sense of Macpherson and Steinhorn [3], [2] as well as 1-based. The argument, involving nonstandard finite cardinalities, is straightforward. A few people asked about this issue in private conversations and communications, in particular Martin Bays Pierre Simon, Dugald Macpherson Charles Steinhorn (in MSRI, spring 2014), and more recently Alex Kruckman. So we thought it worthwhile to clarify the situation with a quick proof. Thanks to all the above people for discussions. Recall the basic notions. A structure M is in language L is pseudofinite if every sentence true in M is true in some finite L-structure. Equivalently M is elementarily equivalent to an ultraproduct of finite L-structures. If M is pseudofinite and saturated say, then every definable set X in M has a “nonstandard finite cardinality” |X| which is an element of a saturated elementary extension of (N,+,×, <, ....), and the map taking X to |X| satisfies the usual properties inherited from the finite setting. Suppose D = M is strongly minimal and saturated. D is said to be unimodular if whenever a = (a1, .., an) and b = (b1, .., bn) are each independent ntuiples from D and a ∈ acl(b) (so also b ∈ acl(a)) then mlt(a/b) = mlt(b/a). Definable means possibly with parameters. We refer to [5] for basics of stability, Morley rank (RM(−)) etc.