This is my first post so I will try to be as clear and concise as possible. I am doing a course in statistics and we define the true mean of a random gaussian variable to be as follows:

$\mu$ = $\int_{R}xdF_{N(\mu, \sigma^2)}(x)$

As the Lesbesgue-Stieljes integral.

In the notes we defined the empirical CDF as follows:

${F_{n}}$ = $\frac{1}{n}\sum_{i=1}^{n}\mathbb{1}_{\{X_{i} \le x \}}$

We then proceed to show that:

$\mu$ = $\int_{R}xd(\frac{1}{n}\sum_{i=1}^{n}\mathbb{1}_{\{X_{i} \le x \}})$ = $\frac{1}{n}\sum_{i=1}^{n}X_{i}$

I am aware that this makes perfect sense as an empirical estimator for the mean, however I am quite confused as to how about deducing this from the integral. We are integrating with respect to a differential of the ECDF, and deduce that the expression is $X_{i}$?

I am struggling to even deduce what the result of $\int_{R}\mathbb{1}_{\{Xi \le x\}}$ evaluates to.

Any help on the above would be much appreciated.