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The monograph contains the detailed exposition of the results obtained by theauthor during the last several years. In particular it contains an improvementof the well known Calderon - Ryff interpolation theorem and description of"verifying" operators of conditional mathematical expectations {if such anoperator leaves an order ideal in $L^1% invariant, then the ideal is aninterpolation space}.

Conditional expectations and interpolation of linear operators on ordered ideals between $L^1(0,1)$ and $L^\infty(0,1)$