I'm trying to find the proof of the following proposition.

Let $\displaystyle S^n$ be the unit sphere in $\displaystyle R^n$. If $\displaystyle K \subset S^n$ is $\displaystyle n-1$ dimensional, then K has Lebesgue measure zero.

I'd b very thankful if any of you could give me the general idea behind the proof or the link to a textbook or internet source where the proof of this is explained.