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Fall 2019 - Problem 4

measure theory

Let $A \subseteq \R$ be measurable with positive Lebesgue measure. Prove that the set $A - A = \set{z - y \mid z, y \in A}$ has non-empty interior.

Hint: Consider the function $\phi\p{x} = \int \chi_A\p{x + y} \chi_A\p{y} \,\diff{y}$ , where $\chi_A$ is the characteristic function of $A$ .

Solution.

By intersecting $A$ with a large enough ball, we may assume without loss of generality that $A$ has finite positive measure. Observe that

\abs{\phi\p{x} - \phi\p{0}} \leq \int_\R \abs{\chi_A\p{x + y} - \chi_A\p{y}} \,\diff{y}.

Since $\chi_A \in L^1$ and translation is continuous in $L^1$ , we see that $\phi$ is continuous at $x = 0$ . Hence, there exists $\delta > 0$ such that if $\abs{x} < \delta$ , then

\phi\p{x} = \int \chi_A\p{x + y} \chi_A\p{y} \,\diff{y} \geq \frac{\phi\p{0}}{2} = \frac{m\p{A}}{2} > 0.

In particular, $\chi_A\p{x + y}\chi_A\p{y} > 0$ , i.e., there exists $y \in A$ such that $x + y \in A$ , and so $x = \p{x + y} - y \in A - A$ for all $\abs{x} < \delta$ . It follows that $B\p{0, \delta} \subseteq A - A$ , so the set has non-empty interior.