On averaging of rearrangement invariant ideals of the space \(L^{1} (\Omega , \Sigma , \mu )\) by non-atomic \(\sigma \)-subalgebras of \(\Sigma \)
Consider a probability space \((\Omega,\Sigma,\mu)\) and a rearrangement invariant ideal \(X\) such that \(L^\infty(\mu) \subseteq X \subseteq L^1(\mu)\). An averaging operator is a conditional expectation operator \(E(\cdot| \mathcal A)\), where \(\mathcal A\) denotes a \(\sigma\)-subalgebra of \(\Sigma\). The image \(E(X| \mathcal A)\) of \(X\) by \(E(\cdot| \mathcal A)\) is a vector ideal. Then it is said that \(\mathcal A\) averages \(X\) if the inclusion \(E(X| \mathcal A) \subseteq X\) holds. The following question is the motivation of the paper. Under which condition does a \(\sigma\)-subalgebra average a rearrangement invariant ideal \(X\)? In particular, the paper deals with the case of non-atomic \(\sigma\)-subalgebras of non-atomic probability spaces (the atomic case is already known). A majorant ideal \(X\) is the one that is averaged by every purely atomic \(\sigma\)-subalgebra. A \(\sigma\)-subalgebra is verifying if it averages only majorant ideals among all rearrangement invariant ideals. After some preliminary results concerning equimeasurability, rearrangement invariance and the majorant property, the main result in Section 5 states that a non atomic \(\sigma\)-subalgebra \(\mathcal A\) is verifying if and only if, for each rearrangement invariant ideal \(X\), the consecutive \(\mathcal A\)-averaging of \(L^1(\mu)\)-rearrangement invariant hull of \(E(X| \mathcal A)\) can be extended to the majorant hull of \(X\) (Theorem 5.3). Finally, Section 6 is devoted to the case of Lebesgue probability spaces, and provides some examples comparing averagings with respect to atomic \(\sigma\)-subalgebras and non atomic ones.
scientific article; zbMATH DE number 5770267