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# The Steinhaus property and Haar null sets

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The Steinhaus property and Haar null sets Marc-Aurele Massard ENS Lyon Contents Introduction 1 1 Preliminaries 1 1.1 Polish group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Haar-null sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Hereditary, dense G? sets and measures 3 2.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 The proof of the theorem . . . . . . . . . . . . . . . . . . . . . . 4 3 Consequences of the theorem 5 4 The Haar-null sets 5 Introduction The purpose is to find a satifactory extension in Polish Groups of the following theorem : Theorem (Steinhaus theorem). Let µ be a translation-invariant regular mea- sure defined on the Borel sets of R, and A is a Borel measurable set with µ(A) > 0, then 0 ? Int(A?A).

• dense g?

• exists y0

• ?k ?

• gh ? ?

• closed nowhere dense

• also let

• b2 ?

• y0

• there exists

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