Let a quantum system be described by Hilbert space $\mathscr{H}$ and let $|\psi\rangle$ be an arbitrary state. Define the operator
$$P=|\psi\rangle\langle \psi|$$
This is hermitian. It has two eigenvalues: $0$ and $1$ with two eigenspaces. The $1$ eigenspace is the subspace spanned by $|\psi\rangle$, in other words $$\mathscr{H}_1=\{\lambda |\psi\rangle : \lambda \in \mathbb{C}\}$$
while the eigenspace corresponding to zero is its orthogonal complement $\mathscr{H}_2 = \mathscr{H}_1^\perp$.
Since this is one observable, one would expect it could be measured. But how physically such measurement can be made?
The point is that $P$ doesn't correspond directly to a physical quantity like momentum, energy or angular momentum, which one experimentalist would know of a procedure to measure in the lab.
The point is that if $A$ is one physical quantity with eigenspaces $\mathscr{H}_\lambda$ corresponding to the values $\lambda\in \sigma(A)$ the postulates of quantum mechanics allows us to say "well the system's state lies in $\mathscr{H}_\lambda$" if when we measure $A$ we get $\lambda$.
This in particular allows us to preparate a system in any eigenstate of any physical quantity that we can measure. But preparing on arbitrary states still is somewhat weird to me.
Of course, if measuring $P$ is possible, a measure of $P$ yielding value $1$ would prepare a system in the state $|\psi\rangle$.
So, is there any "generalized way" to measure this observable?