I'm reading "Ergodic Theory and Semisimple Groups" by Zimmer and at the very beginning of Chapter $2$ (pp. $8$) the author claims that
An action with quasi-invariant measure can be thought of as an action with an invariant measure class.
I interpreted this vague statement in the following way:
Every quasi-invariant measure is in the same measure class with an invariant measure.
Question1: is this statement true? I don't see how to prove this fact.
Question2: If question1 has a negative answer, how should such a statement be understood?
Here the author assumes the group $G$ be locally compact second countable, the action on a standard Borel space $S$ (i.e. Borel isomorphic to a Borel subset of a Polish space) be Borel (i.e. measurable). Moreover, a $\sigma$-finite measure $\mu$ is said to be quasi-invariant under the action of $G$ iff for all $A\subseteq S$, $g\in G$ we have $\mu(Ag)=0\iff\mu(A)=0$. It is invariant iff $\mu(Ag)=\mu(A)$ for all $A$, $g$. Finally, two measures are said to be in the same measure class iff they have the same null sets.
About my background: I have attended a basic measure theory course mostly focused on the real case. Whenever possible, a good reference that covers these topics is appreciated.
Thank you in advance for your help.