Q:

Inequality for open set measures

How do you prove that, given a sequence of open sets $I_i$ (in $\mathbb{R}^n)$, it is possible, given $\varepsilon>0$, to find a s sequence of open sets $J_i$ such that $I_i \subset J_i$ (strict inclusion) and $\sum
|J_i| \leqslant \sum |I_i| + \varepsilon$ ?  Here $|A|$ is the Lebesgue measure of $A$. 

A:

For each $I_i$, pick any $x \notin I_i$. Let $B_i$ be any ball centered at $x$ with measure less than $\frac{\epsilon}{2^{i}}$. Let $J_i = I_i \cup B_i$. $J_i$ is a strict superset since it contains $x$. It is clear that the $|J_i| \leq |I_i| + \frac{\epsilon}{2^i}$.

