I am reading through Lee's Introduction to smooth manifolds and he defines a domain of integration to be a bounded subset of $\mathbb{R}^n$ whose boundary has measure zero. He then goes on to say 'However, since we cannot guarantee that arbitrary open subsets or arbitrary compact subsets are domains of integration...'
I can see why an arbitrary open subset would not necessarily be a domain of integration (doesn't have to be bounded) - but how is it possible to have a compact set which is not a domain of integration? It is bounded and closed, and I'm trying to think of any compact set in, say, $\mathbb{R}^3$, I would think its boundary is 2 dimensional and thus has 3-dimensional measure zero?