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Abstract. Let X be an uncountable Polish space, and let / be a proper fj-ideal of subsets of X such that {x} £ I for each x e X. Denote by Ba(I), a < a>\ , the Baire system generated by the family of functions / : I-tR continuous / almost everywhere. We prove that if r(I) = min{a < oi\ : Ba+i(I) = Ba(I)} , then either r{I) = 1 or r(I) = a)¡ . This answers the problem raised by R. D. Mauldin in 1973.

Abstract. Let X be an uncountable Polish space, and let / be a proper fj-ideal of subsets of X such that {x} £ I for each x e X. Denote by Ba(I), a < a>\ , the Baire system generated by the family of functions / : I-tR continuous / almost everywhere. We prove that if r(I) = min{a < oi\ : Ba+i(I) = Ba(I)} , then either r{I) = 1 or r(I) = a)¡ . This answers the problem raised by R. D. Mauldin in 1973.