Question About Lebesgue measure in $\mathbb{R}^2$.

Question

I'm sitting on a problem form Stein's Real analysis book. He begins by defining sets $\mathcal{O_n} = \{x \in \mathbb{R}^d \colon d(x,E) < 1/n\}$, where $E$ is some measure subset of $\mathbb{R}^d$; he asks us to find an example of an unbounded closed set for which $m(E)$ fails to be the limit of the measure of the $\mathcal{O}_n$. One such example I'm playing around with is $\mathbb{R}$ as a subset of $\mathbb{R}^2$ of measure zero. Pictorially I can see that the measure of each $\mathcal{O_n}$ will be infinite, however, I'm not sure of a way to make it rigorous. Any hints or suggestions would be very appreciated!

Answer 1

You may try with e.g. $E=\{ (x,y) : -1< y(1+x^2)<1 \}$ (for $E$ open).

If $E$ is supposed closed then your example ${\Bbb R}\times\{0\}$ works fine.The $1/n$-neighborhood is a uniform band ${\Bbb R} \times (-1/n,1/n)$ of infinite measure, or else e.g. change to non-strict inequalities in the above example.