This is an problem from real analysis, when learning Riemann-Stieltjes integral:
A discrete random variable is an $X$ such that $X(\Omega)$ is countable (possibly finite), say $X(\Omega) = \left\{x_i, i\in \Bbb N\right\}$. Suppose that these values $x_i$ all lie in $[a,b]$, and suppose that the probability that $X$ takes the value $x_i$ is $p_i \in [0, 1]$, with sum of all $p_i = 1$. The expectation of $X$ is defined to be $$E[X] = \sum_{i\in \Bbb N} x_i p_i$$ Find $F : [a,b] \to [0,1]$, monotone increasing with $F(a) = 0$ and $F(b) = 1$, such that $$E[X] = \int_a^b x \,dF(x)$$
I though I can can construct $F$ as $$\sum_{i\in\Bbb N} p_i I(x-x_i)$$ where $I$ is the left-continuous unit step function, and by the theorems in Rudin, $E[X]$ indeed equals to that integral. Also, $F(a) = 0$ is satisfied. But the only problem is that if $\{x_i, i \in \Bbb N\}$ contains $b$, then $F(b)$ will not equals $1$ (because the left-continuous unit step function $I(x)$ takes $0$ when $x = 0$). So I am struggling how to handle this case.