I find two description about that we have a sequence of discrete probability distribution or measures as follows.
(1): Consider a sequence of discrete probability distribution $P=(P_i)_{I=1}^n$ on $R^d$ such that $P_i\ge 0$ for all $i$ and $\sum_i P_i=1$.
(2): Suppose that $\{x_1,\dots, x_n\}$ is a set of distinct data points taking values in $R^d$. Let $\mu=\sum P_i\delta_{x_i}$.
For (1) and (2), are they saying the same thing? I mean here $P_i$ is the same in (1) and (2), can we say $P=\mu$ ?