Answer to a question of Rosłanowski and Shelah

Abstract

Rosłanowski and Shelah [Small-large subgroups of the reals, Math. Slov. 68(3) (2018) 473–484] asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with [Formula: see text] that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper, we give affirmative answers for the general case.

Small-large subgroups of locally compact Abelian Polish groups

“…every F σ subset of 2 ω ×2 ω which contains a non-null rectangle must contain a measurable Haar-positive rectangle). Then(12) holds in every non-discrete locally compact Polish group G. It is consistent with ZF C (i.e. it holds in the Cohen model) that in every locally compact Polish group meager subgroups are null.…”

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In [13] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely whether it is consistent with ZF C that in every locally compact group there are no meager but non-null subgroups. The answer is affirmative for both questions [12], however, in this paper we present a much simpler proof for the special case of Abelian Polish groups.

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“…The following can be found in [12], but for the sake of completeness we include the proofs. Friedman's theorem can be found in [5]:…”

“…µ). Assume that condition(12) holds in 2 ω (i.e. every F σ subset of 2 ω ×2 ω which contains a non-null rectangle must contain a measurable Haar-positive rectangle).…”