A pseudometric on $\mathcal{M}(X,\mathcal{A})$ induced by a measure

For a probability measure space $(X,\mathcal{A},\mu )$, we define a pseudometric $\delta$ on the ring $\mathcal{M}(X,\mathcal{A})$ of real-valued measurable functions on $X$ as $\delta (f,g)=\mu (X\setminus Z(f-g))$ and denote the topological space induced by $\delta$ as ${\mathcal{M}}_{\delta }$. We examine several topological properties, such as connectedness, compactness, Lindelöfness, separability and second countability of this pseudometric space. We realise that the space is connected if and only if $\mu$ is a non-atomic measure and we explicitly describe the components in ${\mathcal{M}}_{\delta }$, for any choice of measure. We also deduce that ${\mathcal{M}}_{\delta }$ is zero-dimensional if and only if $\mu$ is purely atomic. We define $\mu$ to be bounded away from zero, if every non-zero measurable set has measure greater than some constant. We establish several conditions equivalent to $\mu$ being bounded away from zero. For instance, $\mu$ is bounded away from zero if and only if ${\mathcal{M}}_{\delta }$ is a locally compact space. We conclude this article by describing the structure of compact sets and Lindelöf sets in ${\mathcal{M}}_{\delta }$.