Empirical Measure

Why

There is a natural probability measure on a measurable space to associate with a dataset from the base set of that space.

Definition

The empirical measure for a dataset in some measurable space is the measure which associates to each event the proportion of the records which are elements of that event.

Notation

Let $(a^1,\dots ,a^n)$ be a dataset in a measurable space $(A,\mathcal{A})$. Let $P:\mathcal{P}(A)\to [0,1]$ be the probability measure that assigns to each set $B\subset A$ the number

$$P(B)=\frac{1}{n}|\{k\in \{1,\dots ,n\}|a^k\in B\}|.$$