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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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##### Existential quantification

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The Steinhaus property and Haar null sets
Marc-Aur`eleMassard ENS Lyon
Contents Introduction 1 1 Preliminaries1 1.1 Polishgroup .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.2 Topologicalspaces .. . . . . . . . . . . . . . . . . . . . . . . . .2 1.3 Haar-nullsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 2 Hereditary,denseGδsets and measures3 2.1 Preparation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 2.2 Theproof of the theorem. . . . . . . . . . . . . . . . . . . . . .4 3 Consequencesof the theorem5 4 TheHaar-null sets5
Introduction The purpose is to ﬁnd a satifactory extension in Polish Groups of the following theorem : Theorem(Steinhaus theorem).Letµbe a translation-invariant regular mea-sure deﬁned on the Borel sets ofR, andAis a Borel measurable set with µ(A)>0, then0Int(AA). We just must ﬁnd on which set we would apply the result.In fact it will be on non generically left Haar-null sets.But what is a left Haar-null set ?Let’s start with (a lot of) deﬁnitions.
1 Preliminaries 1.1 Polishgroup Let’s start with some deﬁnition on Polish groups. Deﬁnition(Polish space).A topological spaceXis said to be aPolish space if it is separable and completely metrizable. Deﬁnition(Polish group).A groupGis said to be aPolish groupif it is a topological group which is also a Polish space.
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