Steinhaus theorem

Remarks

One proof uses inner regularity to find a compact set $K \subseteq A$ with $0 < \lambda(K) < \infty$. The function $f(x) = \lambda(K \cap (K + x))$ is continuous and $f(0) = \lambda(K) > 0$, so $f(x) > 0$ for $x$ near $0$. If $f(x) > 0$, then some $k \in K$ also lies in $K + x$, so $x = k - k'$ with $k, k' \in K$, and $x \in A - A$. Example: if $A \subseteq \R$ is a measurable set of positive measure, then every real number $t$ with $\abs{t}$ small enough is a difference of two points of $A$. The hypothesis $\lambda(A) > 0$ is needed: $A = \Z$ has measure zero, and $\Z - \Z = \Z$ contains no interval around $0$. The same statement holds in every locally compact group for a measurable set of positive Haar measure, with $A A^{-1}$ in place of $A - A$.

Going deeper

For a student

The theorem is a key step in automatic continuity results: for instance, a Lebesgue measurable additive function $f \colon \R \to \R$ is continuous, hence linear. It also shows that a measurable proper subgroup of $\R$ has measure zero, since a subgroup containing an interval around $0$ is all of $\R$. Its analogue for Baire category is the Pettis lemma.