In chapter 7 of Probability Essentials (Jacod, Protter), p. 40, distribution functions F are defined as $F(x) = P((-\infty,x])$ and then three characterizing properties are given for a distribution function on $(\mathbb{R}, \mathcal{B})$, where $\mathcal{B}$ is the Borel sigma-algebra on $\mathbb{R}$:
- $F$ is non-decreasing;
- $F$ is right-continuous;
- $\lim_{x\to -\infty} F(x) = 0$ and $\lim_{x\to +\infty} F(x) = 1$
Then the authors go on to prove that any distribution function has those three properties ($\Rightarrow$) and, vice versa, any function satisfying those properties is a distribution function ($\Leftarrow$).
The proof for ($\Leftarrow$) is quite technical. A set function is defined as $$P(A) \equiv \sum_{1\le i \le n} \{F(y_i) - F(x_i)\}$$ for any A in the set $\mathcal{B}_0$ of finite disjoint unions of intervals of the form $(x,y]$, with $ -\infty \le x \lt y \le +\infty $, which is an algebra. If we can prove that $P$ is a probability, we will have proven that F is a distribution function.
The most complicated part is proving countable additivity for $P$ within the algebra $\mathcal{B}_0$, as one can see in book.
My question is, would it not be easier if we defined $P$ on the set $\mathcal{B}_1$ of countable disjoint unions of intervals of the form $(x,y]$, with $ -\infty \le x \lt y \le +\infty $ ? In this way, any countable union of sets $A_i \in \mathcal{B}_1$ is in $\mathcal{B}_1$ and defining $P$ as $$P(\cup_{i=1}^\infty A_i) = \sum_{i=1}^\infty P(A_i) $$ trivially yields countable additivity within $\mathcal{B}_1$. Why are we not allowed to do that? What do we lose?
That would make everything harder. For example, if $[a,b)$ is the union of finitely many disjoint intervals $[a_j,b_j)$ it's easy to show that $P$ is finitely additive by showing that $F(b)-F(a)=\sum(F(b_j)-F(a_j))$; proving even finite additivity is much harder iif we alllow infinitely many intervals.