TITLE: To show that the set point distant by 1 of a compact set has Lebesgue measure $0$
QUESTION [38 upvotes]: Could any one tell me how to solve this one?
Let $K$ be a compact subset of $\mathbb{R}^n$, and $$A:=\{x\in\mathbb{R}^n:d(x,K)=1\}.$$
Show that $A$ has Lebesgue measure $0$.
Thank you!

REPLY [10 votes]: I was going to make this a comment but it occurred to me there might be sufficient interest that perhaps I should not bury it in a comment.
At the beginning of the paper below Erdős gives a short proof (that he attributes to Tibor Radó) making use of the Lebesgue density theorem that $E_r$ has Lebesgue measure zero, where $E$ is a closed set in ${\mathbb R}^n$ and $$E_r \, = \; \{ x \in {\mathbb R}^n : \; d(x,E)=r \} $$
Paul Erdős, Some remarks on the measurability of certain sets, Bulletin of the American Mathematical Society 51 #10 (October 1945), 728-731.
Later in this paper (item 6), Erdős proves the stronger result that, for $K$ compact, the Hausdorff $(n-1)$-measure of $K_r$ is finite, and hence $E_r$ has $\sigma$-finite Hausdorff $(n-1)$-measure when $E$ is closed.
More precise results can be found in a 1985 paper by Oleksiv/Pesin Zbl 573.28010 [English translation: Mathematical Notes 37 (1985), 237-242], and I'm sure there are quite a few related results in the literature. For instance, each of the sets $E_r$ is $[1]$-very porous in the sense defined in this conference talk of mine, and among other things I gave a short argument there that each such set has $\sigma$-finite packing $(n-1)$-measure.
The analysis of results related to these in infinite dimensional normed spaces has also generated a fair amount of interest. One possible entry point into this is Ludek Zajicek's 1983 paper Differentiability of the distance function and points of multi-valuedness of the metric projection in Banach space.
(Added 12 Weeks Later) Someone recently gave me a vote on this answer and, in looking at what I wrote, it occurred to me that a much better reference for the comment "one possible entry point into this" that I made in the last paragraph is the following book:
Joram Lindenstrauss, David Preiss, and Jaroslav Tišer,
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces,
Annals of Mathematics Studies #179, Princeton University Press, 2012, x + 425 pages. Zbl 1241.26001 (a review)
Princeton University Press web page for the book
Amazon.com web page for the book