Let $a,b \in \mathbb{R}$ with $a<b$. Let $$f(x) = \begin{cases} \dfrac{1}{b-a} & \text{if $a<x<b$} \\ 0 & \text{if $x\leq a$ or $x \geq b$} \end{cases}$$ Define $F: \mathbb{R} \rightarrow \mathbb{R}$ by $$F(x) = \int_{-\infty}^x f(t) dt$$Show that the Lebesgue-Stieltjes measure $\mu_F$ associated to F is a probability measure on $(\mathbb{R},B_\mathbb{R})$
So I was able to show that $\mu_F(\mathbb{R}) = F(\infty) - F(-\infty) = \int_a^b \frac{1}{b-a}dt = 1$
And $\mu(\emptyset) = \mu((-\infty,\infty)^c) = \mu(\mathbb{R}) - \mu((-\infty,\infty)) = 1-1 = 0$
What I'm struggling with is to show countable additivity. I defined a sequence of intervals $(a_i,b_i]_{i=1}^{\infty}$ disjoint with $a_1 < b_1 \leq a_2 < b_2 \leq \cdots$. I'm not really sure how to convert $\mu(\bigcup (a_i,b_i])$ into $\sum \mu((a_i,b_i])$. Everything where $b_i < a$ should just be $0$ but after that I'm getting tripped up