Let us define a pure vector state of a quantum system as a vector $\psi$ in a Hilbert space $\mathscr{H}$ with norm $\|\psi\| = 1$. Let $\mathscr{B}(\mathscr{H})$ be the Banach space of bounded linear operators on $\mathscr{H}$.


In the $C^{\*}$-algebra formulation of quantum mechanics, one defines observables as the $C^{\*}$-algebra $\mathscr{B}(\mathscr{H})$ of bounded linear operators on $\mathscr{H}$, and a state as a norm one positive functional $\omega$ on this $C^{\*}$-algebra. Strocchi gives a nice interpretation in his book on why these states should be interpreted as average values of observables. In addition, if $A \in \mathscr{B}(\mathscr{H})$ is fixed, with spectrum $\sigma(A)$ and $C^{\*}(A)$ is the unit $C^{\*}$-algebra generated by $A$, then by Riesz-Markov Representation Theorem, to each state $\omega$ there exists a measure $\mu\_{\omega,A}$ such that:
$$\omega(A) = \int\_{\sigma(A)}\tilde{A}(\lambda)d\mu\_{\omega,A}(\lambda)$$
where $\tilde{A}$ is the Gelfand transform of $A$. If the state is $\omega(A) = \langle \psi, A\psi\rangle$, there is a natural interpretation of $\langle \psi, A\psi\rangle$ as the average value of $A$.


In non-algebraic approaches, one usually postulates that given an observable $A$ and a pure vector state $\psi$, the probability that $A$ will take any value inside a Borel set $E \subset \sigma(A)$ is the spectral measure:
$$p\_{\psi,A}(E) = \langle \psi, \chi\_{E}(A)\psi\rangle$$
where $\chi\_{E}$ is the characteristic function of the set $E$.


My question is: is there a nice interpretation or justification of this definition? I know that, a posteriori, this leads to the various formulas that physicists use, but I wanted something a priori. Why is this natural? Putting it another way, I understand the spectral measure is a probability measure on $\sigma(A)$, but why should I use it as the probability of obtaining $E$ in the state $\psi$? Notice that, in this case, $A$ can be unbounded too.



Interpretation of spectral measures in quantum mechanics
(Small correction: We can take the observables to be the *self-adjoint* elements of $B(H)$, or any C${}^\*$-algebra, and in your whole discussion $A$ should be assumed self-adjoint.)


This can be reduced to the following more fundamental principle. Let $v$ be a unit vector in some Hilbert space $H$ and let $E$ be a closed subspace of $H$. Write $H = E \oplus E^\perp$ and decompose $v = v\_1 + v\_2$ accordingly; then the probability the state represented by $v$ will be measured to belong to $E$ is $\|v\_1\|^2$, the probability it will be measured to belong to $E^\perp$ is $\|v\_2\|^2$.


Now if you have an observable (self-adjoint operator) $A$ with finite spectrum, in particular if $H$ is finite dimensional, then we can decompose $H$ as the orthogonal sum of eigenspaces of $A$, $H = E\_1 \oplus \cdots \oplus E\_n$. If the corresponding eigenvalues are $\lambda\_1, \ldots, \lambda\_n$, then the probability a state represented by a unit vector $v$ will be measured to be in $E\_i$ is $\|v\_i\|^2$ where $v\_i$ is the component of $v$ in $E\_i$, and so the expected value of $A$ for $v$ is $\lambda\_1\|v\_1\|^2 + \cdots + \lambda\_n\|v\_n\|^2 = \langle Av,v\rangle$.


That is, the expected value of $A$ for $v$ is the sum of (value of $A$ on $E\_i$)(probability $v$ belongs to $E\_i$). Make sense?


In the general (infinite dimensional) case you can norm approximate any self-adjoint operator with a self-adjoint operator with finite range, so we can argue by approximation that $\langle Av,v\rangle$ is still the expected value.


