Approximation of analytic by Borel sets and definable countable chain conditions

Abstract

LetI be a σ-ideal on a Polish space such that each set fromI is contained in a Borel set fromI. We say thatI fails to fulfil theΣ ¹ ₁ countable chain condition if there is aΣ ¹ ₁ equivalence relation with uncountably many equivalence classes none of which is inI. Assuming definable determinacy, we show that if the family of Borel sets fromI is definable in the codes of Borel sets, then eachΣ ¹ ₁ set is equal to a Borel set modulo a set fromI iffI fulfils theΣ ¹ ₁ countable chain condition. Further we characterize the σ-idealsI generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property forΣ ¹ ₁ sets mentioned above. It turns out that they are exactly of the formMGR(F)={A : ∀F ∈ F A ∩F is meager inF} 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 onR, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.