Suppose that $X$ is a Banach space and $\mathcal{F}(X) = \overline{span}(\{ \delta_x : x \in X \})$, where $\delta_X : Lip(X) \rightarrow \mathbb{R}$ and $Lip(X)$ is the set of real-valued Lipschitz functions on $X$.
Define $\beta : \mathcal{F}(X) \rightarrow X$ given by $$\beta(\mu) = \int_X x d\mu(x)$$
In this paper, before Lemma $2.4$, the author mentioned that $\beta(\mu)$ is the barycenter of measure $\mu$.
Question: What is the meaning of barycenter of measure $\mu$ and why the integral gives us the barycenter of measure $\mu$?