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 $\mu$?
It generalizes the barycenter of point sets. Namely, if you have a finite set $\left\{p_1,\ldots,p_n\right\}$ and you look at the average of the Dirac measure $\delta_{p_i}$ $$\mu=\frac{\delta_{p_1}+\ldots+\delta_{p_n}}{n},$$ then obvioulsy $\beta(\mu)$ is the barycenter of $\left\{p_1,\ldots,p_n\right\}$.