Explain the proof of proposition (3.4) of Gerald B. Folland "A Course in Abstract Harmonic Analysis" book

Question

Please explain the specified phrase in more details:

Answer 1

If $v \in M^\bot$, then $Pv=0$, since $P$ is the orthogonal projection on $M$.

Further, Proposition (3.1) shows that $M^\bot$ is invariant under each $\pi(x)$. Hence, $\pi(x)v \in M^\bot$. With the same argument as above, we get $P \pi(x)v=0$.