A Fubini theorem

Let I 0 be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I 0 and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak–Rosłanowski–Shelah stating that I 0 on 2 N strongly violates the countable chain condition.