Given a Borel space $\Omega$ and a Hilbert space $\mathcal{H}$.
Consider a spectral measure $E:\mathcal{B}(\Omega)\to\mathcal{B}(\mathcal{H})$.
A spectral measure can be completed $\overline{E}$.
(For details see: Birman & Solomyak, Section 5.2, Page 126-127)
Now, it is crucially for many instances that its probability measures are complete then, too.
But this is to much to hope for...
So what can happen in worst case?
