The theorem can also be extended to nonmeagre sets with the Baire property. The proof of these extensions, sometimes also called Steinhaus theorem, is almost identical to the one below.
Proofedit
The following simple proof can be found in a collection of problems by late professor H.M. Martirosian from the Yerevan State University, Armenia (Russian).
Let's keep in mind that for any , there exists an open set , so that and . As a consequence, for a given , we can find an appropriate interval so that taking just an appropriate part of positive measure of the set we can assume that , and that .
Now assume that , where . We'll show that there are common points in the sets and . Otherwise . But since , and
,
we would get , which contradicts the initial property of the set. Hence, since , when , it follows immediately that , what we needed to establish.
Steinhaus, Hugo (1920). "Sur les distances des points dans les ensembles de mesure positive" (PDF). Fund. Math. (in French). 1: 93–104. doi:10.4064/fm-1-1-93-104..
Weil, André (1940). L'intégration dans les groupes topologiques et ses applications. Hermann.
Stromberg, K. (1972). "An Elementary Proof of Steinhaus's Theorem". Proceedings of the American Mathematical Society. 36 (1): 308. doi:10.2307/2039082. JSTOR 2039082.
Sadhukhan, Arpan (2020). "An Alternative Proof of Steinhaus's Theorem". American Mathematical Monthly. 127 (4): 330. arXiv:1903.07139. doi:10.1080/00029890.2020.1711693. S2CID 84845966.
Väth, Martin (2002). Integration theory: a second course. World Scientific. ISBN 981-238-115-5.