A measurable map with analytic domain and metrizable range is quotient

The aim of this note is to prove the statement in the title which is the natural generalization of the classical theorem of N. Lusin for separable metrizable spaces; for historical remarks and classical proof see K. Kuratowski [9, §28]. If P is a topological space we let Baire (P) denote the set P endowed with the <7-algebra of all Baire sets in P . Recall that the collection of Baire sets in P is the smallest cr-algebra of sets such that each real valued continuous function is measurable. A mapping ƒ :P—>Q of topological spaces is called Baire measurable or simply measurable, if jf:Baire(P)—»Baire (Q) is measurable. A mapping ƒ :P—*(? of measurable spaces is called quotient if ƒ is surjective measurable mapping such that XQQ is measurable if jT""" [-X"] is measurable. Now we are prepared to state our main result; the reader may also read an interesting corollary in Theorem 9 below.