The countable partition averaging operator with respect to a minimal rearrangement invariant ideal of the space $L^1(0,1)$

In terms of functions $f^*$ and $f^{**}$ the necessary and sufficient conditions are given for the validity of the inclusion $\mathsf E(N_f|\mathscr T)\subset N_f$ where $f$ is an arbitrary element of $L^1(0,1)$, $N_f$, $f$, $\mathscr T$ is the minimal rearrangement invariant ideal of $L^1(0,1)$ containing $f$, $\mathscr T$ is a partition of the segment [0,1] by points of a sequence $t_n\downarrow0$ and $\mathsf E(\cdot|\mathscr T)$ is the conditional expectation operator.