Carathéodory's extension theorem to extend CDF $F_X$ to a probability measure $\mathbb{P}_X$

Question

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1. Let $\mathcal{A} = \{ (a, b] : a \leq b \}$. $\mathcal{A} \subseteq 2^\mathbb{R}$. This is a semiring or ring? More details here ...
2. Define a pre-measure $\mu : \mathcal{A} \mapsto [0, 1]$ by $$\mu\left( (a, b ] \right) = F_X(b) - F_X(a)$$
3. Consider the sequence $ \delta_i = (-i, i] \in \mathcal{A}$. Then, $\bigcup_{i=1}^\infty \delta_i = \mathbb{R}$ and $\mu(\delta_i) < \infty$.

Then, under some more conditions, $\mu$ can be extended to a unique probability measure $\mathbb{P}_X$ on the Borel $\sigma$-algebra generated by $\mathcal{A}$.