Abstract
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.
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.
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