Stein's real analysis book, chapter 6, Exercise 15

Question

This is the same question as Infinite product of probability measures is a premeasure but there is one step I don't understand: if $(C_k)_{k=1}^\infty$ is a decreasing sequence in $\mathcal C$ such that $C_k \in \mathcal C_k$ for all $k$, and $\bigcap_{k=1}^\infty C_k = \varnothing$, then $m(C_k) \to 0$. I don't understand why this is sufficient to prove $\sigma$-additivity given finite additivity. I've been stuck on this step for two days, any help is appreciated.

Answer 1

That's a standard fact about finite measures: If $(E_n)_{n}$ are disjoint and measurable, then $A_n:= \bigcup_{k=1}^n E_n$ is increasing and $C_n := A \setminus A_n$ decreasing to the emptyset, where $A= \bigcup_{n=1}^\infty E_n$. Thus $m(C_n) \rightarrow 0$, but $$\mu(C_k) = \mu(A) - \mu(A_n) = \mu(A) - \sum_{k=1}^n \mu(E_k).$$ At this step, we need that $\mu$ is finite and finite additive. All in all, we have $$\sum_{k=1}^n \mu(E_k) \rightarrow \mu(A).$$