20.10 The Fréchet-Nikodym Metric
Definition 20.10.1 (Fréchet-Nikodym Metric).label Let $(X, \cm, \mu)$ be a finite measure space, then the Fréchet-Nikodym metric with respect to $\mu$ is the mapping
\[d: \cm \times \cm \to [0, \infty) \quad (A, B) \mapsto \mu(E \Delta B)\]
For each $A, B \in \cm$, $A$ and $B$ are essentially equal if $\mu(E \Delta B) = 0$. The set $\cm$ modulo essential equality is a complete metric space.
Proof. By the completeness of $L^{1}(X, \cm, \mu; \real)$.$\square$
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