Singular symmetric functionals and stabilizing subspaces of Marcinkiewicz spaces

We consider Marcinkiewicz spaces of functions measurable on a semiaxis that admit a wide set of singular symmetric Dixmier functionals. For elements of these spaces we study the measurability property introduced by A. Connes. We establish that this property is closely connected with the Tauberian property (which is more strong) but is not reduced to it. We specify the maximal subspace of the Marcinkiewicz space such that for its elements both properties are equivalent. We prove that this subspace is not reducible to other known subspaces of the Marcinkiewicz space and that it plays an important role in the theory of Dixmier functionals.