# Lebesgue measure of $\operatorname{Cl}A$ if $\operatorname{Int}A=\emptyset$

Let $A \subset \mathbb{R}$ be a boundary set ($\operatorname{Int}A=\emptyset$) and Lebesgue measure $0$ ($\lambda (A) = 0$). What can we say about Lebesgue measure of $\operatorname{Cl}A$ ?

It is obviously that we have $\lambda \left( \operatorname{Cl}A \right) \ge 0$. But what we can say about the second inequality?

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The rationals have something to say... – David Mitra Jan 1 '14 at 11:27

## 1 Answer

Note that $\operatorname{Int}(\mathbb{Q}\cap[0,1])=\varnothing$, though $\lambda(\operatorname{Cl}(\mathbb{Q}\cap[0,1]))=\lambda([0,1])=1$

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So if $\operatorname{Int}A = \emptyset$ then $\lambda (\operatorname{Cl}A) = \lambda(A)$ ? – Thomas Jan 1 '14 at 11:38
No, this is not true even for my example. Look carefully! – Norbert Jan 1 '14 at 13:15
Yes, my mistake. I don't have any idea how can me help yours answer. I understand what have you wrote, but I don't know how can I use it to find bounded of Lebesgue'a measure of set $A$. Could you give me next hint? – Thomas Jan 1 '14 at 13:35
I do not understand thi sentence "how can I use it to find bounded of Lebesgue'a measure of set $A$" – Norbert Jan 1 '14 at 14:08
I mean $\operatorname{Cl}A$. Because I would like find $a$ such that $\lambda(\operatorname{Cl}A) \le a$. I don't know how can I use your answer to get it $a$. – Thomas Jan 1 '14 at 14:20