= POVM on the occupied support
{c}
{title2=$\sum_rM_r^\dagger M_r=P$}
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 $M_r^\dagger M_r$ form a <POVM>; the operators themselves additionally specify state updates.
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