[Authors]
 A. A. Mekler

[Title]
 Regular functions and conditional expectation operators on ordered ideals
 of $L^1(0,1)$-space

[AMS Subj-class]
 46E30 Spaces of measurable functions

[Abstract]
 The article treats the role of action of conditional expectation operators
 in the real interpolation between $L^1(0,1)$ and $L^\infty(0,1)$. The main
 results are the following:
 1). If all the above mentioned operators translate into itself \Big(one says
     \emph{average}\Big) an real vector ordeded ideal $X$ which lies between
     $L^1(0,1)$ and $L^\infty(0,1)$, then $X$ is interpolated.
 2). There exist a class of sub-$\sigma$-algebras \Big(which are called
     \emph{verifying}\Big) such that averaging of $X$ by an arbitrary verifying
     $\sigma$-subalgebras guarantees for $X$ to be being interpolated where $X$
     is every \emph{symmetrical} \Big(=rearrangement invariant\Big) ideal.
 3). The full classification of verifying sub-$\sigma$-algebras in the class
     of independently complemented sub-$\sigma$-algebras is given.
 4). For a \emph{principal} symmetrical ideal $X$ \Big(=generated by a function
     $f\in L^1(0,1)$\Big) an efficiently criterion is given in order to $X$ were
     averaged by an arbitrary independently complemented sub-$\sigma$-algebra.

[Comments]
 LaTeX, Russian, 85 pp.

[Contact e-mail]
 alexandre.mekler@mail.ru