\end{aligned}$$ which can be rewritten as $$\begin{aligned} \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}^0\circ T_{{\pi}\mu}\, d(q\mu^0)\, = \int_{\Omega} \log{\pi}\circ T_{{\pi}\mu}\, d(q\mu),  \label{eq:theorem0_00}\end{aligned}$$ where $