= Stone formula
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For a <self-adjoint operator> $A$, fix the <resolvent> convention $R(z)=(A-zI)^{-1}$. For real $a<b$, <Stone formula> states the strong-operator limit
$$
\lim_{\epsilon\downarrow0}\frac1{2\pi i}\int_a^b[R(t+i\epsilon)-R(t-i\epsilon)]\,dt
=E_A((a,b))+\tfrac12E_A(\{a\})+\tfrac12E_A(\{b\}).
$$
Equivalently the integrand is $\pi^{-1}\operatorname{Im}R(t+i\epsilon)$. The endpoint halves are necessary when there are eigenvalues at the endpoints; if neither endpoint has a spectral atom, the limit is simply the interval <spectral projection>.<Mathematical Methods in Quantum Mechanics by Gerald Teschl>[first edition, §4.1, Theorem4.3, printed p114]{ref}
To see the limit, use the <spectral theorem> to replace the resolvent difference by the scalar kernel
$$
F_\epsilon(\lambda)=\frac1\pi\int_a^b\frac{\epsilon}{(\lambda-t)^2+\epsilon^2}\,dt.
$$
It lies in $[0,1]$. Its arctangent primitive tends to one inside $(a,b)$, zero outside $[a,b]$, and one half at each endpoint. For every vector $x$, <dominated convergence> applied to $\int|F_\epsilon-F_0|^2\,d\mu_x$ proves convergence in norm after applying the operators to $x$. This explains both the strong rather than necessarily operator-norm limit and the half weights.
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