Consider $[0,1] $ with $\mu$ Lebsgue measure on $[0,1]$ .Let $H = L^2([0,1], \mu)$. For $\phi \in L^{\infty}([0,1],\mu)$.
Define $M_{\phi} \in B(L^2([0,1], \mu))$ by $M_{\phi}(f) = \phi f, \ f\in L^2([0,1], \mu)$ .
I have shown that $\sigma(M_{\phi}) = \text{ess range} \ \phi$.
Now to find the spectral measure $E : M_{\phi} \to B(H)$ of $M_{\phi}$.
Required Hints to do the problem.