# Measure Theory - Why doesn't empty interior imply zero measure?

I'm studying for a qualifying exam in measure theory and ended up "proving" a set with empty interior implies zero measure. I know this isn't true (irrational numbers provide a counterexample in $$\mathbb{R}$$) but can't find my error:

Let $$E \subset \mathbb{R}^n$$ be a set with empty interior. Note that the closure, $$\overline{E} = \text{int}(E) \cup \text{Bd} (E)$$, where $$\text{int}(E)$$ is the interior of $$E$$ and $$\text{Bd}(E)$$ is the set of boundary points of $$E$$. By monotonicity and subadditivity of outer measure, $$|E|_e \le |\overline{E}_e| \le |\text{int}(E)|_e + |\text{Bd}(E)|_e=0$$ by assumption and because the boundary of $$E$$ is $$(n-1)-$$dimensional.

Can anyone find my mistake?

• Essentially, the topological boundary is a very diffuse thing and only very loosely related to the boundary you would encounter in (differential) geometry, where the claim about the dimension would be correct. Jun 7, 2023 at 7:39
• In general, when faced with a possible proof and a possible counterexample to the same proposition, you can apply your proof to your counterexample and see where things break. If $n=1$ and $E$ is the set of irrationals, what are $\operatorname{int}(E)$ and $\operatorname{Bd}(E)$ and $\bar{E}$, what are their measures, and where does a mistake appear in the proof?
– Stef
Jun 7, 2023 at 8:07

You have no reason to assume that the measure of the boundary is $$0$$ too. And you provided an example yourself: the irrationals. Its boundary is $$\Bbb R$$, whose Lebesgue measure is infinity.
As your example with irrational numbers shows, $$\text{Bd}$$ is not necessary $$(n - 1)$$-dimensional, it can very well be the entire space.
• What the sentence in parenthesis means is, for an arbitrary subset, such as $\text{Bd}(E)$ here, it may not be well-defined to talk about what the "dimension" of the subset is. Unless you specify precisely the definition of "dimension" you use. (Of course, if the set is a linear or affine subspace, or an embedded manifold, or something, it is clear what the dimension is.) Jun 7, 2023 at 23:21