functional analysis – How do I show that null sets of a spectral measure are null sets on the induced complex measure?

I’m going through my lecture notes again for functional analysis, and I came up on this property that I can’t seem to prove. So $E(omega)$ is your garden-variety spectral measure, and we define another complex-valued measure $mu_{x,y}(omega) = langle x , E(omega)y rangle$. The notes state that any $E$-null set $omega$ is also a $|mu_{x,y}|$-null set, but I don’t know how to show this directly. It’s clear that $omega$ is a $mu_{x,y}$-null set, but I haven’t been able to move this to the absolute value.

I also couldn’t find a name for this measure $mu_{x,y}$; if it does have a formal name, that would also be great to know!