Lebesgue–Banach points of functions in symmetric spaces

L. Kh. Poritskaya

Matematičeskie zametki, Tome 23 (1978) no. 4, pp. 581-592

Résumé

For a symmetric space $E$ (Ref. Zh. Mat. IIB391) of measurable functions in the interval $[0,1]$ we introduce a characteristic $$ \Pi(E)=\inf\biggl\|\sum_{i=1}^nx_i\biggl(\frac{t-\tau_{i-1}}{\tau_i-\tau_{i-1}}\biggr)\varkappa_{[\tau_{i-1},\tau_i]}(t)\biggr\|_E, $$ where $\varkappa_{[\tau_{i-1},\tau_i]}(t)$ is a characteristic function and the $\inf$ is taken over all $n$ and the sets $x_i(t)\in E$, $\|x_i\|_E=1$ and $\tau_i\in[0,1]$ ($0=\tau_0<\tau_1<\dots<\tau_n=1$, $i=1,2,\dots,n$). We prove the following THEOREM. The conditions $\Pi(E)>0$ and separability are necessary and sufficient for almost all the points of the interval $[0,1]$ to be Lebesgue–Banach points for any function $f\in E$. If at least one of these conditions is not satisfied, then there exists in $E$ a function such that almost all the points of the interval $[0,1]$ are not its Lebesgue–Banach points.