Spectral projection ($E_A(S)=1_S(A)$)

For a self-adjoint operator $A$, its projection-valued spectral measure of a normal operator assigns each Borel set $S\subseteq \mathbb{R}$ an orthogonal projection $E_A(S)$. For a normalized quantum state $\psi$, the probability of a measurement result in $S$ is $⟨\psi ,E_A(S)\psi ⟩$. In finite dimensions, $E_A(S)$ is the sum of the orthogonal projections onto eigenspaces whose eigenvalues lie in $S$. This generalizes an individual eigenvector probability to degenerate eigenvalues and continuous spectra. The functional-calculus notation $1_S(A)$ uses the indicator function of the set.