Example

— Let $\mu$ be a discrete measure on K, positive and of total mass 1; it is thus of the form $\mu = \sum_{i=1}^n \lambda_i \varepsilon_{x_i}$, where $x_i \in K$, and the $\lambda_i$ are real numbers such that $\lambda_i \geq 0$ for all $i$ and $\sum_{i=1}^n \lambda_i = 1$. Since $\int x\, d\varepsilon_y(x) = y$ (Ch. III, §3, No. 1, Example 3), $b_\mu = \int x\, d\mu(x) = \sum_i \lambda_i x_i$. In particular, for every $x \in K$, x is the barycenter of the measure $\varepsilon_x$.