Geodesic
A~supplement to the paper ``The countable partition averaging operator with respect to a~minimal rearrangement invariant ideal of the space $L_1(0,1)$''
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 137-141

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Let $\mathcal A$ be a countable partition of $[0,1]$ whose elements have positive measure. For $f\in L_1(0,1)$ the symbol $N_f$ denotes the smallest rearrangement invariant ideal sublattice of $L_1(0,1)$ containing $f$. Conditions are given under which $E(N_f|\mathcal A)\subset N_g$ for some $g\in L_1(0,1)$. It is also stated that $E(f|\mathcal A)\prec 2^5E(f^*|\mathcal A^*)$, where $\prec$ is the Hardy–Littlewood preorder on $L_1(0, 1)$ and $\mathcal A^*$ is a decreasing rearrangement of $\mathcal A$. 
@article{ZNSL_1986_149_a11,
     author = {A. A. Mekler},
     title = {A~supplement to the paper {``The} countable partition averaging operator with respect to a~minimal rearrangement invariant ideal of the space $L_1(0,1)$''},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {137--141},
     publisher = {mathdoc},
     volume = {149},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a11/}
}
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A. A. Mekler. A~supplement to the paper ``The countable partition averaging operator with respect to a~minimal rearrangement invariant ideal of the space $L_1(0,1)$''. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 137-141. http://geodesic.mathdoc.fr/item/ZNSL_1986_149_a11/