As I know the spectral measure of a linear possibly unbounded self adjoint operator $T$ on a seperable Hilbert space is a projection valued measure defined by its measurable functional calculus $A\mapsto \chi_A(T)$.
But it seems like sometimes $\mu_{x}(A):=\langle P_Ax,x\rangle$ is also called spectral measure, where $\langle \cdot,\cdot \rangle$ is the dot product in the Hilbert space.
isn't that ambiguous? I would also like to get to know the jargon of spectral theory correctly and since I'm learning spectral theory on my own, I'd like to ask if it's common to call $\mu_x$ also spectral measure?