Non-existence of space with measure with following property

Question

I need to show that there is no such space $(\Omega , \Sigma , \mu)$ for which Every self-adjoint operator unitary equivalent to multiplication operator in $L_2 (\Omega , \mu)$. I think I just need some hints on attempting this problem. Any ideas?

Answer 1

Consider a self-adjoint operator $A$ with empty point spectrum. Show that if $A$ is unitarily equivalent to a multiplication operator on $L^2(\Omega,\mu)$, then $(\Omega, \mu)$ contains no atoms.

Consider a rank-one orthogonal projection operator $P$. Show that if $P$ is unitarily equivalent to a multiplication operator on $L^2(\Omega,\mu)$, then $(\Omega, \mu)$ contains an atom.