A supplement to the paper "The countable partition averaging operator with respect to a minimal rearrangement invariant ideal of the space $L_1(0,1)$"

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 $\mathcal A$.