I am learning Lebesgue–Stieltjes integration and decided to test myself computing the expected value of a 6-sided die roll "from scratch".
I know to obtain the expected value I need to compute:
$$\mathbb E(x) = \int x \,dg(x)$$
and I know $g(x)$ is the Cumulative Distribution Function:
$$\begin{equation*} g(x) = \begin{cases} 0 & x \in (-\infty, 1) \\ \frac{1}{6} & x \in [1, 2) \\ \frac{2}{6} & x \in [2, 3) \\ & \vdots \\ 1 & x \in [6, \infty) \end{cases} \end{equation*}$$
But I am stuck there. My questions are:
what is $x$ in this case? The outcome of the die roll? Its density function?
can we say that the integrand function is simple? In the case of simple functions I know we can use this formula:
$${\displaystyle \int s\,\mathrm {d} \mu _{w}=\sum _{i}a_{i}\mu _{w}(A_{i})} $$
so the result of the integral will be:
$$\frac{1}{6} \cdot a_1 + \frac{2}{6} \cdot a_2 + \dots + 1 \cdot a_6 $$
but I do not know what $a_i$s are.
So my question is: what is the step by step method of calculating $\mathbb E(die)$ with Lebesgue-Stieltjes integral?