I am currently studying Durrett's Probability Theory and Examples.
I have question with this excercise.
$A$ is almost invariant if $P(A\Delta \varphi^{-1}(A))=0$ and call $C$ invariant in the strict sense if $C=\varphi^{-1}C$. Let $A$ be any set. Prove $A$ is almost invariant if and only if there is a $C$ invariant in the strict sense with $P(A\Delta C)=0$.
Proving the if part, in the solution, the following equality is used. $$P(\varphi^{-1}(A)\Delta C) = P(\varphi^{-1}(A)\Delta \varphi^{-1}(C)) =P(\varphi^{-1} (A\Delta C))=P(A\Delta C)=0$$
However how does $P(\varphi^{-1}(A\Delta C))= P(A\Delta C)$ hold?
Doesn't the above equality hold when $\varphi^{-1}(A-C)=A-C$ which is not mentioned in the problem?