Symmetric Difference On Measure Spaces
As long as there is a notion of "how big" a set is, the symmetric difference between two sets can be considered a measure of how "far apart" they are. Formally, if μ is a σ-finite measure defined on a σ-algebra Σ, the function,
is a pseudometric on Σ. d becomes a metric if Σ is considered modulo the equivalence relation X ~ Y if and only if . The resulting metric space is separable if and only if L2(μ) is separable.
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