1

Consider a compact interval $\mathcal{I}$ of the real line and two non empty sets $A$ and $B$ that partition $\mathcal{I}$ (so $A$ and $B$ are disjoint and their union is $\mathcal{I}$).

Assume that $B$ is a closed set and has at least a limit point (= accumulation point = cluster point).

Moreover, $\forall x,y\in B$, where $x<y$, it holds $(z,w)\subset(x,y)$, where $(z,w)\subset A$ and $z<w$.

Does $B$ have (Lebesgue) measure zero?

Mathgineer94
  • 61
  • 4

1 Answers1

2

So-called "fat" Cantor sets provide counterexamples. See this MSE post for specific examples.

Lee Mosher
  • 135,265
  • Thanks! So in general I need more info on the measure of the intervals $(z,w)$? – Mathgineer94 Feb 16 '23 at 18:41
  • Yeah. The measure of $B$ is the measure of $\mathcal I$ minus the measure of $\mathcal I-B$, and $\mathcal I-B$ is a disjoint union of countably many intervals $I_1 \cup I_2 \cup \cdots$, so the measure of $\mathcal I-B$ is just the sum of the infinite series $\sum_{k \ge 1} \text{Length}(I_k)$. – Lee Mosher Feb 16 '23 at 21:18