Why is the sample mean the expectation of the empirical distribution?

Question

I came across the empirical distribution $$\hat P(A) = \frac{1}{n}\sum \mathbf{1}_{\{X_i \in A\}}$$today, whose cdf is given by $$\hat F(x) = \frac{1}{n}\sum \mathbf{1}_{\{X_i \leq x\}}. $$ The proof on the lecture notes was $$\int xd\hat P(X) = \frac 1 n \sum \int xd\delta_{X_i} = \frac 1 n \sum X_i = \bar X $$ I am having a hard time understanding the second-to-last equality.
I know the definition of $\delta_{X_i}$, which is just $\delta_{X_i}(A) = \mathbf{1}_{\{X_i \in A\}}$. But I cannot understand beyond this.
Any help is appreciated. Thank you.

Answer 1

There $\delta _{X_i}$ is the Dirac measure at $X_i$ (a random measure, as $X_i$ is a random variable), therefore $$ \int f\, d \delta_{X_i}=f(X_i) $$ for any measurable function $f$. Taking $f$ as the identity function (usually represented by $\operatorname{id}$ or just $x$) you get the RHS.