Hi I'm reading Ambrosio : Gradient Flows in Metric Spaces 2nd Edition.
Let $X$ be a Polish space and let $\mathcal{P}(X)$ be Borel probability measures on $X$, analogously define $\mathcal{P}(X\times X)$. Let $\gamma \in \mathcal{P}(X\times X)$ have 1st marginal $\mu\in \mathcal{P}(X)$, and admit the following disintegration w.r.t to $\mu$ :
$$ \gamma=\int_X \gamma_{x_1}d\mu(x_1). $$
Then Ambrosio defines (Page 126 definition 5.4.2) the barycentric projection $\overline{\gamma}:X \to X$ as
$$\overline{\gamma}(x_1)=\int_X x_2 d\gamma_{x_1}(x_2).$$
I'm REALLY struggling to see the meaning of this projection/what role it plays. To me this projection just maps an element in $X$ to "the $\gamma$ conditional distribution (conditioned that the first element is $x_1$?) ". Does anyone have some more knowledge about how to view this projection ?