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2016 Topological linear subspace of $L_0(\Omega, \mu)$ for the infinite measure $\mu$
Yoshiaki Okazaki
Nihonkai Math. J. 27(1-2): 147-154 (2016).


Let $(\Omega, \mathcal{A}, \mu)$ be a measure space. We shall characterize the maximal topological linear subspace $M_{\infty}$ of $L_0(\Omega, \mathcal{A}, \mu)$ in the case where $\mu(\Omega)=+\infty$. $M_{\infty}$ is the truncated $L_{\infty}$ space which is open and closed in $L_0(\Omega, \mathcal{A}, \mu)$. In the case where $\Omega=\textbf{N}$(natural numbers), $\mu(A)=\sharp A=$ the cardinal number of $A$, the maximal linear subspace of $L_0(\textbf{N}, \mu)$ is $\ell_{\infty}$.

Funding Statement

This work is based on research 26400155 supported by Grant-in-Aid for Scientific Research (C) from Japan Society for the Promotion of Science.


Download Citation

Yoshiaki Okazaki. "Topological linear subspace of $L_0(\Omega, \mu)$ for the infinite measure $\mu$." Nihonkai Math. J. 27 (1-2) 147 - 154, 2016.


Received: 1 January 2016; Revised: 7 September 2016; Published: 2016
First available in Project Euclid: 14 September 2017

zbMATH: 06820454
MathSciNet: MR3698248

Primary: 46A16 , 46E30
Secondary: 28A20

Keywords: $L_0$ , convergence in measure , measurable function , topological linear space , truncated $L_{\infty}$ space

Rights: Copyright © 2016 Niigata University, Department of Mathematics

Vol.27 • No. 1-2 • 2016
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