  We show that a partition of the unity (or POVM) on a Hilbert space that is
almost orthogonal is close to an orthogonal partition of unity in the same von
Neumann algebra. This generalizes to infinite dimension and slightly
strengthens previous results in matrix algebras by Kempe-Vidick and
Ji-Natarajan-Vidick-Wright-Yuen.
  We also generalize to infinite dimension a duality result between POVMs and
minimal majorants of finite subsets in the predual of a von Neumann algebra.
