In an unpublished paper by Ackerman-Freer-Patel (Forum Math. 2016; arXiv link), it is stated that if a relational structure has trivial definable closure, it has an invariant measure.

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 say? It's 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?