What are measurable physical quantities? The book "Fundamentals of many body physics" by Wolfgang Nolting at the beginning of chapter 3 says:

Measurable physical quantities are:

*

*the eigenvalues of observables

*the expectation values of observables $\langle\hat{A}(t)\rangle, \langle\hat{B}(t)\rangle, ...$

*the correlation functions between observables $\langle\hat{A}(t)\cdot\hat{B}(t)\rangle...$

I've taken quantum mechanics and quantum field theory courses, but it was my understanding that measurable quantities are only the eigenvalues of observables.
Why and how are (2) and (3) measurable?
 A: You probably just mean something different by "measurable" than what Nolting has in mind here.
Recall that the very nature of an expectation value - indeed, why it is the thing we "expect" - is that if you measure the corresponding observable (i.e. "do a measurement" that results in an eigenvalue of the observable) on an identically prepared state over and over, then in the limit of infinite measurements the average of the measurement results converges to the theoretical expectation value. In this sense an expectation value is "measureable", since you can quantify how well - within experimental errors and the limitation of finitely many measurements - your experimental average compares to this theoretical value.
Likewise, correlation functions often appear in theoretical predictions for experimental results like distributions for the results of scattering experiments, and hence are likewise "measureable" - just not by the usual idea of a single strong measurement of a quantum observable like the eigenvalues of that observable are.
A: I will address this part of the question

but it was my understanding that measurable quantities are only the eigenvalues of observables.

You are right that eigenvalues of observables are the only "measurable" quantities (barring the subtleties of the word "measurable" mentioned by @ACuriousMind), but I would like to highlight that this is far less restrictive than it sounds. You can construct "new" observables from "old", whose eigenvalues are just functions of the eigenvalues of the "old" observables.
The idea behind this statement is that if $f(x)$ is some analytic function, and $A$ is an observable with eigenvalue decomposition $A = \sum_{i}\lambda_{i}P_{i}$ where $\lambda_{i}$ are the eigenvalues of $A$ and $P_{i}$ are the projection operators onto the i'th eigenspace, then $f(A)$ is also an observable, and it has eigenvalue decomposition $f(A) = \sum_{i}f(\lambda_{i})P_{i}$.
In this sense one can imagine constructing various interesting measurable quantities out of some set of simpler observables.
