Problem: Let $A,B\subset \mathbb{R}^{n}$ be bounded and Jordan measurable, and let $R\supset A,B$ be a rectangle. Then $A\cup B$ and $A\cap B$ are Jordan measurable and $$c(A\cup B)+c(A\cap B)=c(A)+c(B).$$
Our definition is that a set $A$ is Jordan measurable if the numbers
$$\underline{c}(A)=\sup_{P\in\mathcal{P}}\left\{\underline{J}(P,A)\right\}$$
$$\overline{c}(A)=\inf_{P\in\mathcal{P}}\left\{\overline{J}(P,A)\right\}$$
coincide, where
$$\underline{J}(P,A)=\sum\left\{m(I_{k})\ \vert\ I_{k} \mathrm{\ interval\ of\ }P\mathrm{\ contained\ in \ int}(A)\right\}$$
$$\overline{J}(P,A)=\sum\left\{m(I_{k})\ \vert\ I_{k} \mathrm{\ interval\ of\ }P\mathrm{\ intersecting\ } \overline{A}=A\cup \partial A\right\}$$
and $\mathcal{P}$ is the set of finite partitions of $R$.
I am not sure if I am approaching this the correct way. I wanted to use the fact that if $C, D$ are Jordan measurable with $D\subset C$, then $C\setminus D$ is Jordan measurable with $c(C\setminus D)=c(C)-c(D)$ and apply this to $\partial (A\cup B)\subset \partial A\cup \partial B$ since $\partial A=\partial B=0$, but I realized that I cannot assume that $\partial A\cup \partial B$ is Jordan measurable with $c(\partial A\cup \partial B)=0$.
The only other thing I considered is to take four different collections of intervals in $P$ intersecting $\mathrm{int}(A)$, $A\cup\partial A$, $\mathrm{int}(B)$, and $B\cup \partial B$ respectively, and try to come up with an equality to show $\underline{J}(P,A\cup B)=\overline{J}(P,A\cup B)$ and $\underline{J}(P,A\cap B)=\overline{J}(P,A\cap B)$, but I thought that this may be an overly complicated approach.
I appreciate any suggestions!
Best Answer
I think your idea using $\partial(A \cup B) \subseteq \partial(A) \cup \partial(B)$ is a good one. It will be easier to show that a union of two sets with content zero has content zero than showing the general proposition. Once you know that a union of two measurable sets or a difference of two comparable measurable sets is measurable, and that areas are additive in the case of a disjoint union, the rest of the problem can be done with formal manipulations only.