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This article is cited in 1 scientific paper (total in 1 paper)
Lebesgue–Banach points of functions in symmetric spaces
L. Kh. Poritskaya Scientific-Research Institute "Gidropribor"
Abstract:
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.
Received: 24.06.1976
Citation:
L. Kh. Poritskaya, “Lebesgue–Banach points of functions in symmetric spaces”, Mat. Zametki, 23:4 (1978), 581–592; Math. Notes, 23:4 (1978), 317–324
Linking options:
https://www.mathnet.ru/eng/mzm8173 https://www.mathnet.ru/eng/mzm/v23/i4/p581
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