Suppose $T$ is a self-adjoint operator in $B(H)$ with $\sigma(T)$ a spectrum of $T$. $\mu$ is a spectral measure. For the operators having a generally continuous spectrum how to calculate the multiplicity function? Where multiplicity is usually called spectral multiplicity. Up to compact operators we know how to decompose the spectrum and get the multiplicity function on each bit of disjoint chunks. But how to apply these in continuous spectral measure case? Is related to the direct integral decomposition of the von Neumann algebra generated by $\pi(C(\sigma(T))$ or not?
If I am not clear, please help me. Thanks in advance.