Let $μ$ be a Caratheodory extension of an elementary measure $μ_0$ defined on a semiring $P$. Let $A$ be a measurable set with $μ(A)<∞$. Show there are sets $B,C∈σ(P)$ such that $B⊂A⊂C$ and $μ(C\setminus B)=0$.
I managed to show the existence of $C$ using the identity:
$μ(A)$=$\inf\{\sum_{j=1}^\infty\mu(R_j):(R_j)_{j=1}^\infty\subset P, A\subset\cup_{j=1}^\infty R_j\}$
Where $μ≡μ_0$ on $P$. I used it to prove that there is a set $C∈σ(P)$ such that $A⊂C$ and $μ(A)=μ(C)$. I believe this is exactly the set $C$ I will need for the exercise. But how to find $B$? The identity I wrote is all about sets that contain $A$, but I don't know anything about the subsets of $A$ itself. I thought about using the fact that Caratheodory measure is complete and about taking the supremum of the measures of subsets of $A$ that are in $σ(P)$, but that didn't help me so far. Any ideas?
Let $S$ be the set of all sequences $(R_j)$ in $P$ which satisfies the assertion on the definition of $\mu(A)$.
Given a natural $n$, there is a sequence $(R_{j,n})_{j\in \mathbb{N}}$ in $S$ which satisfy $$\mu(\bigcup _j R_{j,n} - A)=\sum_j \mu(R_{j,n})-\mu(A) < \frac{1}{n}$$
So, what you say about this set: (?) $$C=\bigcap _{n=1}^{\infty}\bigcup_{j=1}^{\infty}R_{j,n}$$
You can prove that $C \in \sigma(P)$ and $\mu(C\setminus A)=0$.