$A$-Systems, Independent Functions, and Sets Bounded in Spaces of Measurable Functions

S. Ya. Novikov

Matematičeskie zametki, Tome 75 (2004) no. 1, pp. 115-134

Résumé

Let $U\subset L_\circ\bigl([0,1],\mathscr M,\mathbf m\bigr)$ be a set of Lebesgue measurable functions. Suppose also that two seminormed spaces of real number sequences are given: $\mathscr A$ and $\mathscr B$. We study $(\mathscr A,\mathscr B)$ -sets $U$ defined by the classes $\mathscr A$ and $\mathscr B$ as follows: $$ \begin{gathered} \forall a=(a_n)\in\mathscr {A},\quad \forall(f_n(t))\in u^{\mathbb{N}}\quad\text{(or for sequences similar to,} \\ \quad (f_n(t)) \quad\exists E=E(a)\subset[0,1],\quad \mathbf m E=1\quad\text{such that} \\ \{a_nf_n(t)\mathbf{1}_E(t)\}\in\mathscr B,\qquad t\in[0,1]. \end{gathered} $$ We consider three versions of the definition of $(\mathscr A,\mathscr B)$ -sets, one of which is based on functions independent in the probability sense. The case $\mathscr B=l_\infty$ is studied in detail. It is shown that $(\mathscr A,l_\infty)$ -independent sets are sets bounded or order bounded in some well-known function spaces ($L_p$, $L_{p,q}$, etc.) constructed with respect to the Lebesgue measure. A characterization of such sets in terms of seminormed spaces of number sequences is given. The $(l_1,c_\circ)$- and $(\mathscr A,l_1)$ -sets were studied by E. M. Nikishin.