Uniqueness of a locally solid topology on a Riesz space of measurable functions

Let $\Omega$ be a nonempty set and let $\mathcal{A}$ be a $\sigma$-ring of subsets of $\Omega$ containing all singletons. Denote by $L(\mathcal{A})$ the Riesz space of real-valued $\mathcal{A}$-measurable functions on $\Omega$. This space is locally convex-solid when equipped with the topology $\tau_p$ of pointwise convergence on $\Omega$. We present several conditions equivalent to the nonexistence of other Hausdorff locally solid [respectively, locally convex-solid] topologies on $L(\mathcal{A})$. Some of them assert the nonexistence of $\sigma$-order continuous submeasures or two-valued measures on $\mathcal{A}$ vanishing on singletons.