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I was working through an exercise in Heil's "Introduction to Real Analysis" when I reached a point where I was unsure how to proceed past.

The problem

Let $S_r = \{x \in \mathbb{R}^d : ||x|| = r\}$ be the sphere with radius $r$ in $\mathbb{R}^{d}$ centered at the origin. Prove that $|S_{r}| = 0$.

My attempt

We have that the surface area of the $d$-sphere should be $C_{d}c^{d-1}$ where $C_{d}$ is a constant. Naturally then, it makes sense to consider the "unfurled" $d$-sphere which looks like some $d-1$ dimensional surface. Presumably then, one should be able to cover the $d-1$ dimensional surface with $d$ dimensional boxes in an efficient way (i.e. how one can show the measure of the natural embedding of the real line in $\mathbb{R}^{2}$ is measure zero).

My issue, however, is that I have no clue how to formalize this intuition of unfurling. I would appreciate any guidance or alternative hints in this area.

kodiak
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  • The general fact you seem to be trying to prove is that the volume of a submanifold of lower dimension is always zero. That is true and can be done in more or less the way you say, but formalising this without any familiarity with differential topology is probably going to be tough. There are also straightforward analytic proofs (for example, you can argue by induction). – tomasz Sep 14 '23 at 20:28
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    You might argue that the sphere is contained in a (set-theoretic) difference of $d$-balls whose volume is arbitrarily small, compare https://math.stackexchange.com/questions/625 – Andrew D. Hwang Sep 14 '23 at 22:58

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A simple way would be to argue by contradiction.

If $|S_r|=\epsilon>0$, then by scaling properties of Lebesgue measure we have $|S_{\lambda r}|=\epsilon \lambda^d>0$ for every $\lambda>0$. Then the unit ball is a union of uncountably many disjoint measurable sets of positive measure, and therefore has infinite measure. (Note that for this fact we never have to use additivity for an uncountable union, which would of course be problematic. Instead we just use that an uncountable collection of positive numbers has a countable subset summing to $\infty$.)

M W
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