Let $A$ be a bounded self-adjoint operator on a separable Hilbert space $\mathcal{H}$: $$ A\in\mathcal{B}\left(\mathcal{H}\right)\,,\,A=A^\ast$$ Stone's formula (Reed & Simon Theorem VII.13, as an example for one source) gives the following way to compute the projection-valued measures for $A$:
If $\left(a,\,b\right)\in\mathbb{R}^2$ such that $a<b$ and $\chi_S:\mathbb{R}\to\mathbb{R}$ is the characteristic function of the set $S\subseteq\mathbb{R}$ then $$ \frac{1}{2}\left(\chi_{\left(a,\,b\right)}\left(A\right)+\chi_{\left[a,\,b\right]}\left(A\right)\right)=slim_{\varepsilon\to0^+}\frac{1}{2\pi i}\int_a^b\underbrace{ \left[\left(A-x-i\varepsilon\right)^{-1}-\left(A-x+i\varepsilon\right)^{-1}\right]}_{=2 i \varepsilon\left|\left(A-x+i\varepsilon\right)^{-1}\right|^{2}}dx$$
I have two (related) questions:
Now assume that $a\notin\sigma\left(A\right)$ and $b\notin\sigma\left(A\right)$. Then the above formula gives us simply $\chi_{\left(a,\,b\right)}\left(A\right)$. However, what happens, for example, when $a\notin\sigma\left(A\right)$ and $b\in\sigma\left(A\right)$? Is there a way to recover the spectral measure $\chi_{\left(a,\,b\right)}\left(A\right)$ alone, and not only the average, by a similar integral formula on the resolvent? Unless one knows something about the spectrum at the end-points of the interval, what exactly can be done to get the actual spectral measures and not just averages?
What is the usage of this formula for constructive functional calculus, where one actually needs the projection valued measure $$ P:\mathbb{R}\supseteq S\mapsto\chi_{S}\left(A\right)\in\mathcal{P}\left(\mathcal{B}\left(\mathcal{H}\right)\right) $$ for a measurable set $S$ (where $\mathcal{P}\left(\mathcal{B}\left(\mathcal{H}\right)\right)$ is the set of self-adjoint idempotents) and then one can write $$ f\left(A\right) = \int_\mathbb{R}f(x)dP\left(x\right) $$ But how do you (constructively) go from Stone's formula to $P$? By the Lebesgue-Radon-Nikodym theorem (Rudin's RCA Theorem 6.10, for example), which, I assume, holds for projection-valued measures as well, we could write (uniquely) $$ P = P_a + P_s$$ where $P_a$ is absolutely-continuous with respect to the Lebesgue measure and $P_s$ is singular with respect to $\lambda$. Then if it turns out that $P_s=0$ we could write (denoting by $\frac{dP_a}{d\lambda}$ the Radon–Nikodym derivative with respect to $\lambda$) $$ f\left(A\right) = \int_\mathbb{R}f(x)\frac{dP_a}{d\lambda}\left(x\right)d\lambda\left(x\right)$$However, how do you know (from Stone's formula) if $P_s=0$ and if so, how do you actually compute $\frac{dP_a}{d\lambda}$ (from Stone's formula) I guess it should be equal to $$ \frac{dP_a}{d\lambda}\left(x\right) = slim_{\varepsilon\to0^{+}}\frac{\varepsilon}{\pi}\left|\left(A-x+i\varepsilon\right)^{-1}\right|^{2}$$ though I am not sure how to prove this; and if $P_s\neq0$, how do you compute it as well (again, from Stone's formula), and add that contribution to $f\left(A\right)$? I realize $P_s$ should be further decomposed into pure-point and singular continuous parts, the pure-point has an easy functional calculus form $P_{pp}=\sum_{n}x_n v_{n}\otimes v_{n}^{\ast}$ where $\left(v_n\right)_n$ are the eigenvectors of $A$ and $\left(x_n\right)_n$ are the corresponding eigenvalues, which implies $$f_{pp}\left(A\right) = \sum_{n}f\left(x_n\right)v_n\otimes v_n^{\ast}$$ In pp. 271 of Reed & Simon they have a formula saying $$ P_{pp}\left(\{x\}\right) = slim_{\varepsilon\to0}i\varepsilon\left(A-x-i\varepsilon\right)^{-1}$$ (though I am not sure how to prove it) so that we could write $$ f_{pp}\left(A\right) = \sum_{x} f\left(x\right) P_{pp}\left(\{x\}\right) $$; the singular continuous part could be constructed similarly to the last integral above if one had its Radon-Nikodym derivative $\frac{dP_{sc}}{d\mu}$ with respect to some other measure, say $\mu$, with respect to which $P_{sc}$ is supposedly absolutely continuous. Then we could tentatively write $$ f\left(A\right) = \int_\mathbb{R}f(x)\frac{dP_a}{d\lambda}\left(x\right)d\lambda\left(x\right) + \int_\mathbb{R}f(x)\frac{dP_{sc}}{d\mu}\left(x\right)d\mu\left(x\right)+\sum_{x} f\left(x\right) P_{pp}\left(\{x\}\right)$$ So if all of this is correct, how do you get $\mu$, $\frac{dP_{sc}}{d\mu}$, from Stone's formula or some other formula which involves only integrals on resolvents, and how do you prove these two statements about $\frac{P_{ac}}{d\lambda}$ and $P_{pp}$?
The connection with the spectral measure $P$ is $$ P(E) = \chi_{E}(A). $$ So, for example, $$ P[a,b] = \chi_{[a,b]}(A), \;\; P(a,b) = \chi_{(a,b)}(A) \\ P[a,b] = P(a,b) + P\{a\}+P\{b\} $$ The spectral measure $P$ is regular in the strong topology, which gives you \begin{align} P[a,b]x & = \lim_{\epsilon\downarrow 0} P[a-\epsilon,b+\epsilon]x = \lim_{\epsilon\downarrow 0} P(a-\epsilon,b+\epsilon)x \\ P(a,b)x & = \lim_{\epsilon\downarrow 0} P(a+\epsilon,b-\epsilon)x = \lim_{\epsilon\downarrow 0} P[a+\epsilon,b-\epsilon]x \end{align} So once you have $$ \frac{1}{2}\{ P[a,b]x + P(a,b)x \}, $$ you can take strong limits to obtain $P[a,b]x$ and $P(a,b)x$.
If you're dealing with continuous functions, the spectral integral is a Stieltjes type of integral that converges in the strong topology. That is, $$ \left(\int f dP\right)x = \int_{-\infty}^{\infty} f(t)dE(t)x, $$ where $E(t) = P(-\infty,t]$ or $E(t)= P(-\infty,t)$ (which normalization you use is irrelevant for a continuous function $f$.) And you can determine $E(t)$ constructively from Stone's formula, with the added strong limits mentioned in the previous paragraph. For bounded operators, the integral on the right reduces to a finite interval. Then you can extend to more general functions just as you would extend the Riemann-Stieltjes extend to a Lebesgue-Stieltjes integral. One way to do this is to establish the generalized Parseval equality, $$ \left\|\int f dPx\right\|^2 = \int |f(t)|^2d\|E(t)x\|^2. $$ Then you can replace the Stieltjes integral on the right with the Lebesgue-Stieltjes measure obtained from the non-decreasing function $e_x(t)=\|E(t)x\|^2$. This gives a way of lifting from scalar integrals to the operators in the strong topology. There are tedious details to address, but mostly of the critical ideas are reduced to Measure Theory, with a few final details of representing a bounded sesquilinear form by an operator in order to lift from scalar forms to operators. From scalar to vector to operator is the standard procedure.
Measure Decomposition: Stone's formula is related to the Poisson representation for positive harmonic functions on the upper half plane. Let $R(\lambda)=(A-\lambda I)^{-1}$. Then \begin{align} &\frac{1}{2\pi i}(R(u-iv)x-R(u+iv)x,x) \\ & = \frac{1}{2\pi i}\int_{-\infty}^{\infty}\left[\frac{1}{u-iv-t}-\frac{1}{u+iv-t}\right]d(P(t)x,x) \\ & = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{v}{(t-u)^2+v^2}d\mu_{x}(t), \end{align} where $\mu_x(S) = (P(S)x,x)$ is a finite positive Borel measure on $\mathbb{R}$ with $\mu_x(\mathbb{R})=\|x\|^2$. If $\mu$ is a finite positive Borel measure $\mu$ on $\mathbb{R}$, then $$ F(u+iv) = \frac{1}{\pi}\int_{-\infty}^{\infty}\frac{v}{(t-u)^2+v^2}d\mu(t) $$ is a positive harmonic function in the upper half plane. All components of the measure $\mu$--atomic, absolutely continuous, and singular continuous--can be isolated by limits of the function $F$. As you already know, the full measure $\mu$ can be extracted using $$ \lim_{\epsilon\downarrow 0}\int_{a}^{b}F(u+iv)du = \frac{1}{2}\{\mu[a,b]+\mu(a,b)\} $$ The atomic part is extracted using $$ \lim_{\epsilon\downarrow 0}\pi v F(u+iv) = \mu \{u\} $$ The first set of limits exist for all intervals $[a,b]$, and the second limits exist for all $\{u\}$ (they're zero if $\mu \{u\}=0$.) The following limits exist a.e. with respect to Lebesgue measure $m$: $$ \frac{d\mu}{dm} = \lim_{v\downarrow 0}F(u+iv) \;\;\; a.e. [dm]. $$ Here $\frac{d\mu}{dm}$ is the Radon-Nikodym derivative of $\mu$ with respect to $m$. Finally, the singular continuous part of the measure is what remains after subtracting the atomic and absolutely continuous from the full measure $\mu$.
Reference for Measure Decomposition: Peter Duren, Theory of $H^p$ Spaces
Start with Chapter 1, page 1. This material is fundamental to the study of harmonic and analytic functions on the disk. Duren covers this introductory material in a very classical way using the Riemann-Stieltjes integral, and differentiation theorem for monotone functions. The first 10 pages will give you just about everything, and more.