  A consistent functional calculus approach to the spectral theorem for
strongly commuting normal operators on Hilbert spaces is presented. In contrast
to the common approaches using projection-valued measures or multiplication
operators, here the functional calculus is not treated as a subordinate but as
the central concept. Based on five simple axioms for a "measurable functional
calculus", the theory of such calculi is developed in detail, including
spectral theory, uniqueness results and construction principles. Finally, the
functional calculus form of the spectral theorem is stated and proved, with
some proof variants being discussed.
