I'm recently studying some topics related to spectral theory and I found out that one can extend the Borel functional calculus (that is a -* omomorphism from Borel complex valued bounded functions to strong closure of C*-Algebra generated by fixed operator A) to a unique *-isomorphism that is surjective in Strong closure of A (that also is the bicommutant by von Neumann theorem).
But to say that one has to define a basic measure at the beginning. I understood the definition but still don't get which role does it specifically play in the context of L \infty calculus. Can anyone explain me? Thank you!!