In an unpublished paper by Ackerman-Freer-Patel, it is stated that if a relational structure has trivial definable closure, it has an invariant measure. This finding is very new (1 year old) and hence is not published. Here, however, is what they have give so far: http://www.maths.leeds.ac.uk/events/lmsnorth2011/freerpatel.pdf

My question is: What exactly do THEY mean by an invariant measure? If i had to explain to my uneducated (in graph theory) friend what they meant by an invariant measure, what would I tell him? Its obviously not as trivial as same number of edges, or same planarity... or is it?

If I were to guess, I would say a distribution on edge relations that is unchanged under any permutation of the vertices, but even that accounts for trivial invariant measures. Can someone clear this up? What do those three people mean specifically???