Approximation of analytic by Borel sets and definable countable chain conditions

Abstract

Let I be a σ-ideal on a Polish space such that each set from I is contained in a Borel set from I. We say that I fails to fulfil the Σ_1^1 countable chain condition if there is a Σ_1^1 equivalence relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of Borel sets from I is definable in the codes of Borel sets, then each Σ_1^1 set is equal to a Borel set modulo a set from I iff I fulfils the Σ_1^1 countable chain condition. Further we characterize the σ-ideals I generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for Σ_1^1 sets mentioned above. It turns out that they are exactly of the form MGR(F)={A : ∀F ∈ F A ∩F is meager in F} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the σ-ideal of meager sets is the unique σ-ideal on R, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.

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