If $H$ is a infinit dimensional seperable Hilbert space and $A:D(A)\subset H \rightarrow H$ a possibly unbounded selfadjoint operator. Then it is said that
$$A=\int\limits_{\sigma(A)} \lambda dP_{\lbrace \lambda \rbrace}$$
where $\Omega\mapsto P_{\Omega}$ is the spectral measure of $A$.
What does this actually mean? I mean what does it mean to integrate the identity wrt. to a projection valued measure.
Is it just a different notation to
$$\langle Ax,x\rangle= \int\limits_{\sigma(A)} \lambda dE_x(\lambda)$$ where $E_x(\Omega):=\langle x, P_\Omega x\rangle $
If yes, good. If not, how do these notations relate anyways?