We prove measurable analogues of Whitney's classical theorems on weak isomorphisms of finite graphs. In the setting of locally finite graphings, we introduce a notion of weak isomorphism as an edge-measure-preserving Borel bijection that preserves cycles and hyperfinite subgraphs, modulo null sets. We first show a rigidity theorem, proving that for weakly 3-connected infinitely-ended graphings, every weak isomorphism is induced by an isomorphism of graphings. To our knowledge, this gives the first general sufficient condition in measurable combinatorics for the existence of an isomorphism between two given graphings. Next, we give a full measurable version of Whitney's theorem, showing that every weak isomorphism between graphings can be implemented by countably many measurable Whitney operations, which we introduce in this setting. The proofs require new measurable-combinatorial tools, including a careful analysis of infinitely-ended subforests. This work further develops the limit theory of matroids recently initiated by Lovász.