Convergence of conditional means in the norm of the space BMO
The author introduces a class of interval \(\sigma\)-algebras on the segment \([0,1].\) These are \(\sigma\)-algebras generated by a countable or finite family \(\{I_ n\}\) of closed intervals \(I_ n\subset [0,1],\) where \(\cup I_ n=[0,1]\) and \(I^ 0_ n\cap I^ 0_ m=\emptyset,\quad n\neq m,\) \(I^ 0\) being the interior of I. He obtains an estimate of the operator norm of the conditional expectation with respect to the interval \(\sigma\)-algebra, acting in the BMO space. The characterization of the VMO space is given in terms of the convergence in the BMO norm of sequences of conditional means with respect to flows of interval \(\sigma\)-algebras with various properties.
scientific article; zbMATH DE number 4198745