Every set mentioned is a subset of the real numbers.
Let $m^*(C)$ denote the outer measure of a set $C$. Let $E$ be $any$ set and $A,B$ be measurable, disjoint sets. I'm trying to show that $$m^*(E\cap (A\cup B))=m^*(E\cap A)+m^*(E\cap B).$$
Proof: ($\le$) follows by the countable subadditivity of the outer measure since $$E\cap (A\cup B)= (E\cap A)\cup (E\cap B).$$
Here's where I get stuck:
($\ge$) My attempts have reduced to something of the form:
- There are bounded open sets $G_1, G_2$ containing $E\cap A, E\cap B$, respectively, such that $$m(G_1)\ge m^*(E\cap A),\quad m(G_2)\ge m^*(E\cap B).$$ Hence $$m^*(E\cap A)+m^*(E\cap B)\le m(G_1)+m(G_2).$$ And I would like to extend this inequality to $m(G_1\cup G_2)$ but I know that's not even true, especially since the sets $G_1, G_2$ may not even be disjoint.
I also tried de la Vallée-Poussin Criterion: Let $\epsilon>0$. Since $A, B$ are measurable, there are closed subsets $F_1, F_2$ of $A,B$ respectively, such that $m^*(A\cap E - F_1)+ m^*(B\cap E-F_2)<\epsilon$. Even if I could show, $$|m^*(E\cap (A\cup B)-[(F_1\cup F_2))+ m^*(A\cap E - F_1)+ m^*(B\cap E-F_2)]|<\epsilon.$$
I'm not sure what that would mean.
What I know:
Measure has only been defined for bounded sets.
A bounded set $A$ is $measurable$ if its outer and inner measures are equal; if so, the measure of $A$ is the common value of these measures.
Differences, countable unions, countable intersections of measurable sets are measurable.
The union of a set of pairwise disjoint measurable sets is measurable, with the measure of the union equal to the sum of the measures of the sets in the union.
Outer and inner measures are monotone increasing functions.
Countable subadditivity for outer measure, which states that if $A$ is a countable or finite union of sets $A_i$ then $m^*(A)\le \sum m^*(A_i)$.
De la Vallée-Poussin Criterion, which states that a bounded set $A$ is measurable iff for every $\epsilon >0$ there is a closed set $B\subset A$ such that $m^*(A-B)< \epsilon$.
For any bounded set $B$, I can always find a set $C$ that is a countable intersection of open sets for which $B \subset C$ and $m^*(B)=m^*(C)$.
If $A$ and $B$ are measurable sets, then $m(A\cup B) + m(A\cap B) = m(A) + m(B)$.
If $A$ is bounded and $I$ is an open interval containing $E$, then $m^*(E) + m_*(I-E) = m(I)$.