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  "body": "\n## Summary\n\nA projection-valued measure assigns to each measurable set an orthogonal projection on a Hilbert space, in a way that is additive like a measure.\nIt is the object that the spectral theorem attaches to a self-adjoint or normal operator, and it allows one to integrate functions against an operator.\nIn quantum mechanics, projection-valued measures describe sharp measurements: the projection of a set corresponds to the event that the measured value lies in that set.\n\n## Explanations\n\n### L1 human\n\nAn ordinary measure gives each set a number, its size.\nA projection-valued measure gives each set a projection, a way to cut out a piece of the Hilbert space.\nDisjoint sets give perpendicular pieces, and the whole set gives the whole space.\nFor example, on $\\C^2$ with the standard basis $e_1, e_2$, send a subset $B$ of $\\R$ to the projection onto the span of those $e_k$ with $k \\in B$.\nThis splits $\\C^2$ into the pieces attached to the points $1$ and $2$, as the eigenspaces of the matrix $\\operatorname{diag}(1, 2)$ do.\n",
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