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  "body": "\n## Definition\n\nLet $(\\Omega, \\Sigma)$ be a measurable space, so that $\\Sigma$ is a [[sigma-algebra|0SV8K0TDGY]] of subsets of $\\Omega$.\nLet $H$ be a complex [[Hilbert space|HM2HHE3J51]] with identity map $I$.\nA *projection-valued measure* on $(\\Omega, \\Sigma)$ is a map $E$ from $\\Sigma$ to the bounded operators on $H$ such that:\n\n1. $E(B)$ is an [[orthogonal projection|0B8KZRT28M]] for every $B \\in \\Sigma$;\n2. $E(\\emptyset) = 0$ and $E(\\Omega) = I$;\n3. $E(B \\cap C) = E(B) E(C)$ for all $B, C \\in \\Sigma$;\n4. for all $x, y \\in H$, the map $E_{x,y}(B) = \\ip{E(B) x}{y}$ is a [[complex measure|NY6TYHRGMS]] on $\\Sigma$.\n\n## Remarks\n\nClause 4 is equivalent to the strong countable additivity: if $B = \\bigcup_n B_n$ with $B_n \\in \\Sigma$ pairwise disjoint, then $E(B) x = \\sum_n E(B_n) x$ for every $x \\in H$.\nBy clause 3, the projections of disjoint sets have orthogonal images, and all $E(B)$ commute.\nFor each $x$, $E_{x,x}$ is a positive measure of total mass $\\norm{x}^2$.\nExample: let $H = \\ell^2$ with orthonormal basis $(e_n)_{n \\ge 1}$, let $\\Omega = \\R$ with its Borel sets, and let $(\\lambda_n)$ be real numbers.\nLet $E(B)$ be the orthogonal projection onto the closed span of $\\set{e_n : \\lambda_n \\in B}$; clauses 1 to 4 follow from the fact that $E_{x,y}(B) = \\sum_{n : \\lambda_n \\in B} x_n \\overline{y_n}$.\nNon-example: on $\\Omega = \\R$, let $E(B) = P$ if $0 \\in B$ and $E(B) = 0$ otherwise, for an orthogonal projection $P \\ne I$; then $E(\\R) = P \\ne I$, so clause 2 fails.\n",
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