I've read the following proposition in a book on quantum theory.
Proposition. If a quantum system is in a state described by a unit vector $\psi$ and for some quantum observable $\hat{f}$ we have $\hat{f}\psi=\lambda \psi$ for some $\lambda\in \mathbb{R}$, then
(1)$$E(f^m)=\langle \psi, (\hat{f})^m\psi\rangle=\lambda^m$$ for all positive integers $m$.
(2) The unique probability measure consistent with this condition is the one in which $f$ has the definite value $\lambda$, with probability one.
Part (1) is clear to me. To explain and prove (2) the author writes
"...we want to find a probability measure $\mu$ on $\mathbb{R}$ such that $$\int_\mathbb{R} x^m\, d\mu=\lambda^m,$$ for all non-negative integers $m$. The proposition is claiming that there is one and only one such measure, namely the $\delta$-measure at the point $\lambda$.''
I'm not sure how the integral is formed and why it does not explicitly depend on $f$. Prior to reading this paragraph I was thinking that for a function $g$, the expectation value is computed by $E(g)=\int_{\mathbb{R}} g\, d\sigma$ where $\sigma$ is a suitable probability measure.
[Edit: According to the spectral theorem, there exists a unique spectral measure $\mu_{\hat{f}}$ such that $$\hat{f}=\int_{\sigma(\hat{f})}x\, d\mu_{\hat{f}}(x),$$ and $$\langle \phi, \hat{f}\phi\rangle=\int_{\sigma(\hat{f})}x\, d\mu(x),$$ where $\mu(E)=\langle \phi, \mu_{\hat{f}}(E)\phi\rangle$. Thus, I guess the author's explanation of part 2 is implicitly based on the spectral theorem (and functional calculus), although the spectral theorem appears much later in the book. Q: Is there any way to get the integral in question without resorting to the spectral theorem?]
Could someone, please, clarify this confusion about the definition/computation of the expectation value via integration against a probability measure?
I would like to edit my question as follows:
a) Isn't the proposition claiming the uniqueness of a measure $\sigma$ satisfying $E(f)=\int_\mathbb{R} f\, d\sigma$ rather than the uniqueness of $\mu$? (It's not a 100 percent clear to me that these two are equivalent, although I've not thought about it carefully yet.)
b) Since the author proves the uniqueness of $\mu$ (by referring to a theorem about moment problems in probability theory) I wonder if the uniqueness of $\mu$ can be derived from the uniqueness of the spectral measure $\mu_{\hat{f}}$ associated to $\hat{f}$.