A pseudometric on M(X,A) induced by a measure

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