Infinite dimensional analysis

Question

I got completely confused with this theorem .

How could this measure be invariant for this kind of transformation?

I would be very grateful if someone could explain it

Answer 1

Assume 1) holds, then $\mu=\mu\circ\xi^{-1}$, then $\mu\circ T_{\xi}(f)=\mu(T_{\xi}(f))=\mu(f\circ\xi)=\left<f,\mu\circ\xi^{-1}\right>=\left<f,\mu\right>=\mu(f)$, so $\mu=\mu\circ T_{\xi}$, this is the invariant.