According to the Connes-Feldman-Weiss theorem, measurable amenability is equivalent to measurable hyperfiniteness for countable group actions It is easy to see that Borel hyperfiniteness implies Borel amenability, but the opposite direction is a very hard open problem (Jackson-Kechris-Louveau).
The graph-theoretical analogue of amenability is clearly Property A. One can view the so-called uniform local amenability (ULA) property (or local Følner-property) as the graph-theoretical analogue of hyperfiniteness. Brodzky et al. proved that Property A implies ULA. Two years ago I was able to prove that, in fact, ULA is equivalent to Property A. This result has an interesting application in distributed computing.
We say that an action is Uniformly (measurably) Hyperfinite, if all the positive measure subgraphings of the graphing of the action is hyperfinite with the same structure constants. I recently proved that Uniform Hyperfiniteness is equivalent to Uniform Amenability (the measurable version of Property A). Note that UA ⟹ UH is very similar to the Connes-Feldman-Weiss proof, UH ⟹ UA requires some new ideas.