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In this paper first we define generalized Carleson mea- sure. Then weconsider a special case of it, named conditional Carleson measure on theBergman spaces. After that we give a characterization of conditional Carlesonmeasures on Bergman spaces. Moreover, by using this characterization we find anequiv- alent condition to boundedness of weighted conditional expectationoperators on Bergman spaces.

Conditional Carleson measures and related operators on Bergman spaces