It is well known that there are two formulations of the so-called “spectral theorem” or “Borel functional calculus”. The first approach is via the construction of a “spectral measure”, which is described in details in [Murphy, Section 2.5]; the other approach exploits the representation theory of abelian C*-algebras (See [Davidson, II.2]). These two approaches originate from different interpretations of the finite dimensional spectral theorem.
Let 
be a normal matrix in 
, the “resolution of the identity” version of the spectral theorem says that we can write $$T = \sum_{\lambda}\lambda E(\lambda),$$ where 
runs over the eigenvalues of and 
is the orthogonal projection onto the corresponding eigenspace. The spectral measure approach is then seen as an attempt to generalize this picture.
On the other hand, the “unitary diagonalization” version says that there is a unitary matrix 
such that $$UTU^* = \left( \sum_{i=1}^n \lambda_i E_{i,i} \right),$$ where each 
is an eigenvalue and 
is the 
-matrix unit. One can think of the diagonal matrix \( \sum_{i=1}^n\lambda_iE_{i,i}\) as a multiplication operator on 
, where 
is the discrete n-point space 
and 
is the counting measure. Then clearly the representation approach follows this direction. Note that one can obtain a spectral measure as in the general spectral theorem by declaring 
, where 
is a subset of 
(the Borel 
-algebra in this case is the collection of all subsets). Of course the projection 
here is nothing but the orthogonal projection onto the subspace spanned by the elements of .
The exposition above follows largely the wonderful notes written by Jesse Peterson [Peterson].
References:
[Murphy] G. Murphy, C*-algebras and operator theory, Academic Press, 1990.
[Davidson] K. Davidson, C*-Algebras by Example, AMS, 1996.
[Peterson] J. Peterson, Notes on von Neumann algebras. 2013. (link)
