I'm reading the book Computational Optimal Transport, by Peyré and Cuturi. In the book, in Remark 2.19 the authors make a claim regarding the translation of measures. While trying to prove the remark, I was faced with the following identity to be proven:
Let $\gamma$ be a coupling of $(\alpha,\beta)$ and $\gamma’$ be a coupling of $(T_{\tau\#}\alpha,T_{\tau'\#}\beta)$ where $\alpha$ and $\beta$ are probability measures in $X$ and $Y$, respectively. Also, note the following:
- Coupling - It means that the marginal distributions of $\gamma’$ are $T_{\tau\#}\alpha$ for the project on $X$ and $T_{\tau'\#}\beta$ for the projection on $Y$;
- $T_\tau(x) = x-\tau ,\quad T_{\tau'}(y) = y - \tau'$;
- $\int f(x) d T_{\tau\#}\alpha = \int f(T(x)) d\alpha$.
With all this being said, how does one proves that: $$ \int \langle x,y\rangle d\gamma’ = \int\langle x-\tau,y-\tau'\rangle d\gamma $$