Lebesgue outer measure is not finitely additive

Question

I am reading the book "Lebesgue Measure and Integration: An Introduction" by Frank Burk, and it says the following:

If the Lebesuge outer measure is finitiely additive, then it is countably additive.

How can I prove this?

Answer 1

Let $\mu^*$ be an outer measure which is finite additive, let $(A_n)$ be pairwise disjoint. By countable subadditivity we have $$\mu^*\left(\bigcup_n A_n\right)\leq\sum_n\mu^*(A_n)$$.

By monotonicity and finite additivity we have $$\sum_{n=1}^N\mu^*(A_n)=\mu^*\left(\bigcup_{n=1}^NA_n\right)\leq \mu^*\left(\bigcup_{n=1}^\infty A_n\right)，\forall N\in\mathbb{N}$$

Let $N\to\infty$ we have $$\sum_{n=1}^\infty\mu^*(A_n)\leq \mu^*\left(\bigcup_{n=1}^\infty A_n\right)$$