Boundary of the dense set Let $\{x_i\}$ be the sequence of all rational points in $\mathbb R^n$, and let
$$B_i=\{x\in\mathbb R^n\mid\|x-x_i\|\leq2^{-i}\},~~E=\bigcup^\infty_{i=0}B_i.$$
We know that the Lebesgue measure of $E$, denoted by $|E|$, is finite and $E$ is dense in $\mathbb R^n$. 
Q: Does $\partial E=\mathbb R^n\setminus E$ have the infinite Lebesgue measure?  

In the arithmetic of the extended-non-negative-real-half-line $S=[0,\infty]$ we have $x+\infty=\infty +x=\infty$ for all $x\in S$, and $x+y$ has its usual meaning for $x,y\in [0,\infty).$
Let $M_n$ be Lebesgue measure on $\mathbb R^n.$ For any disjoint Lebesgue-measurable $E,F\subset \mathbb R^n$ we have $M_n(E)+M_n(F)=M_n(E\cup F ).$
So $M_n(E)+M_n(\mathbb R^n$ \ $E)=M_n(\mathbb R^n)=\infty.$ So if $M_n(E)<\infty$ then $M_n(\mathbb R^n$ \ $E)$ cannot be finite.
