I was reading Spectrum and dynamics where Paolo Facchi discusses projection-valued measures and integrals. The discussion and constructions are all based on the fact that one can define characteristic functions of self-adjoint operators, i.e. if $A$ is a self-adjoint operator on a Hilbert space $\mathscr{H}$ and $\chi_{\Omega}$ is the characteristic function associated with the Borel set $\Omega \subseteq \mathbb{R}$, then $\chi_{\Omega}(A)$ is a well-defined object. Again, this is used as the starting point for the discussion and, at the begining of the text, the author says that he'll defer the problem of explicitly defining it. However, I've read the entire text and got confused, since I did not find any such definition. So, my question is: how do we define $\chi_{\Omega}(A)$?
Remark: As mentioned in my other question Extension of functional calculus of continuous functions, I know there is this Borel functional calculus where one can define functions of self-adjoint operators. The problem is that if I use the Borel functional calculus to define $\chi_{\Omega}(A)$, the integral formalism discussed in the text (which is what I'm interested in at the moment) becomes kinda cyclic, since it becomes another way to define the Borel functional calculus. So, I was looking for a more practical definition of $\chi_{\Omega}(A)$ which can be used to give meaning to the linked text hypothesis.
