Topological linear subspace of $L_0(\Omega, \mu)$ for the infinite measure $\mu$

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 $\ell_{\infty}$.