Perfect matchings in hyperfinite graphings

We characterize hyperfinite bipartite graphings that admit measurable perfect matchings. In particular, we prove that every regular hyperfinite one-ended bipartite graphing admits a measurable perfect matching. This allows us to extend the Lyons-Nazarov theorem by showing that a bipartite Cayley graph admits a factor of iid perfect matching if and only if the group is not isomorphic to the direct product of $\mathbb{Z}$ and a finite group of odd order, answering a question of Kechris and Marks in the bipartite case. Our theorems may also be applied to construct measurable equidecompositions, which gives a simple approach to measurable circle squaring and an alternative proof of Borel circle squaring. Our approach can be used more generally for rounding measurable flows and fractional perfect matchings.