Let $X$ be a $\sigma$-finite measure space. $M_\phi :L_2(\mu)\rightarrow L_2(\mu)$ for $\phi \in L_\infty (\mu)$ is defined by $f \rightarrow \phi. f$. $f_0$ is called a star cyclic vector for $M_\phi$ if $L_2(\mu)=cl \{ A . f_0 : A \in C^*(M_\phi)\} $. Give a necessary and sufficient condition on $\phi$ for $M_\phi$ to have a cyclic vector.
I have reached at the following condition:
$f_0$ is a star cyclic vector for $M_\phi$ if and only if for all $f \in L_2(\mu)$ there is a polynomial net $p_\lambda$ such that $p_\lambda(\phi).f_0 \rightarrow f$.
Then I got this:
Assume $f_0$ is a star cyclic vector, then there does not exist two sets, $A,B\subset X$ s.t. $A \cap B= \phi $, $\mu(A),\mu(B)>0$ and $\phi(A)=\phi(B)$.
My question is: is it possible from here to conclude that $\phi$ is one one except on a measure zero set? For what $X$ can we conclude that?