The Hahn-Hellinger Theorem

Question

People can tell the question is not up to the mark, but I felt without understanding the Hahn-Hellinger Theorem properly there is no point talking von Neumann algebras for myself. So can any body help me out clearing the concept of Hahn-Hellinger Theorem, what is the game playing inside the theorem, I tried many books and I am not able to get with clarity. More specifically how cyclic vectors are connected with multiplicity of self-adjoint operator, this theorem say any self-adjoint operator in $\mathcal{H}$ is equivalent to a multiplication operator $M_{z}$ on $\oplus L^2(X,\mu_{i})$, how it looks the case at least for finite dimensional case, how direct sum take cares multiplicity of the eigenvalue that I did not get, Also it is not clear all the measure joined to single measure. Further why cyclic decomposition remembers the spectral multiplicity. Please help. Thanks in advance