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Quite a while ago, I heard about a statement in measure theory, that goes as follows:

Let $A \subset \mathbb R^n$ be a Lebesgue-measurable set of positive measure. Then we follow that $A-A = \{ x-y \mid x,y\in A\}$ is a neighborhood of zero, i.e. contains an open ball around zero.

I now got reminded of that statement as I have the homework problem (Kolmogorov, Introductory Real Analysis, p. 268, Problem 5):

Prove that every set of positive measure in the interval $[0,1]$ contains a pair of points whose distance apart is a rational number.

The above statement would obviously prove the homework problem and I would like to prove the more general statement. I think that assuming the opposite and taking a sequence $\{x_n\}$ converging to zero such that none of the elements are contained in $A$, we might be able to define an ascending/descending chain $A_n$ such that the union/intersection is $A$ but the limit of its measures zero. I am in lack of ideas for the definition on those $A_n$.

I am asking specifically not for an answer but a hint on the problem. Especially if my idea turns out to be fruitful for somebody, a notice would be great. Or if another well-known theorem is needed, I surely would want to know. Thank you for your help.

  • 0
    For future readers: for the first point, with respect to $\mathbb{R}^1$ also see https://math.stackexchange.com/questions/1079464/difference-of-elements-from-measurable-set-contains-open-interval2018-02-22

3 Answers 3

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Here's an attempt at a hint for the first result you ask about: Assume without loss of generality that $A$ has finite measure. Let $f$ be the characteristic function of $A$ and let $\tilde{f}$ be the one of $-A$. The convolution $g = f \ast \tilde{f}$ is continuous and $0$ is in the support of $g$.

Added later: One nice standard application is that every measurable homomorphism $\phi: \mathbb{R} \to \mathbb{R}$ is continuous. For more on that and related matters have a look at these two MO-threads:

  1. On measurable homomorphisms $\mathbb{C} \to \mathbb{C}$.
  2. On measurable automorphisms of locally compact groups

They might elucidate what is mentioned in another answer.


Update:

What I wrote above is the way I prefer to prove this.

Another approach is to appeal to regularity of Lebesgue measure $\lambda$ (used in $1$ and $2$ below).

  1. Since $A$ contains a compact set of positive measure, we can assume $A$ to be compact right away (as $B-B \subset A - A$ if $B \subset A$).
  2. There is an open set $U \supset A$ such that $\lambda(U) \lt 2 \lambda(A)$.
  3. Since $A$ is compact there is $I = (-\varepsilon, \varepsilon)^{n}$ such that $A + x \subset U$ for all $x \in I$.
  4. Since $\lambda (U) \lt 2\lambda(A)$ we must have $\lambda((A + x) \cap A) \gt 0$.

This is of course very closely related to the argument given by Chandru1 below.

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    @Peter: Ok, very good. I updated my answer2011-05-14
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Assume $A$ is the set contained in [0,1] with positive measure, say $m(A) > 0$ with $m$ the Lebesgue measure on $[0,1]$. Let Q be the set of all rational numbers in $[0,1]$ Since $\mathbb{Q}$ is countable, it can be presented as

$\mathbb{Q}=\{p_1, p_2, ..., p_n, ...\}$

Let

$A_n = A+ p_n = \{x+p_n\mid x\in A\}.$

If there exists a pair of integers $n$ and $m$ such that $A_n$ and $A_m$ intersect, then the claim of this proposition is proved. If no such pair exists, then the set of $\{A_n\}$ are all disjoint. Since the union of this family of sets is contained in $[0,2]$ and since $m(A) > 0$, we have

$2 = m([0,2]) \geq m( \bigcup A_n ) = \sum\limits_{n \in \mathbb{N}} m(A_n) =\infty\cdot m(A) = \infty .$

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    I was wondering if there is a simpler proof compared to the more general solution of (https://math.stackexchange.com/questions/1079464/difference-of-elements-from-measurable-set-contains-open-interval), and here I found yours. Thanks.2018-07-10
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If you need to find information on the subject, the first proof of the fact that the set of differences contains a neighbourhood of the origin is (for Lebesgue measure on the line) due to Steinhaus. There is a substantial collection of generalizations of the result.