APPROXIMATION OF ANALYTIC BY BOREL SETS AND DEFINABLE COUNTABLE CHAIN-CONDITIONS

Let I be a sigma-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 Sig ma(1)(1) countable chain condition if there is a Sigma(1)(1) equivalen ce relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of B orel sets from I is definable in the codes of Borel sets, then each Si gma(1)(1) set is equal to a Borel set module a set from 1 I iff I fulf ils the Sigma(1)(1) countable chain condition. Further we characterize the sigma-ideals I generated by closed sets that satisfy the countabl e chain condition or, equivalently in this case, the approximation pro perty for Sigma(1)(1) sets mentioned above. It turns out that they are exactly of the form MGR(F) = {A :For All F is an element of FA boolea n AND F is meager in F} for a countable family F of closed sets. In pa rticular, we verify partially a conjecture of Kunen by showing that th e sigma-ideal of meager sets is the unique sigma-ideal on R, or any Po lish group, generated by closed sets which is invariant under translat ions and satisfies the countable chain condition.