State Carathéodory's extension theorem. Define all terms used in the statement.

Let $\mathcal{A}$ be the ring of finite unions of disjoint bounded intervals of the form

$$A=\bigcup_{i=1}^{m}\left(a_{i}, b_{i}\right]$$

where $m \in \mathbb{Z}^{+}$and $a_{1}<b_{1}<\ldots<a_{m}<b_{m}$. Consider the set function $\mu$ defined on $\mathcal{A}$ by

$$\mu(A)=\sum_{i=1}^{m}\left(b_{i}-a_{i}\right)$$

You may assume that $\mu$ is additive. Show that for any decreasing sequence $\left(B_{n}: n \in \mathbb{N}\right)$ in $\mathcal{A}$ with empty intersection we have $\mu\left(B_{n}\right) \rightarrow 0$ as $n \rightarrow \infty$.

Explain how this fact can be used in conjunction with Carathéodory's extension theorem to prove the existence of Lebesgue measure.