 POVM on the occupied support

If every allowed input density operator has support of a positive operator inside a subspace with orthogonal projector $P$, completeness need only equal $P$ on that subspace. To define a measurement on the full Hilbert space, one can append $M_{\perp }=I-P$ when the original operators act only on the subspace. The added outcome has zero probability on supported inputs. The effects $M_r^{†}{M}_r$ form a POVM; the operators themselves additionally specify state updates.