The Space of Signed Measures
The sum of two finite signed measures is a finite signed measure, as is the product of a finite signed measure by a real number: they are closed under linear combination. It follows that the set of finite signed measures on a measure space (X, Σ) is a real vector space; this is in contrast to positive measures, which are only closed under conical combination, and thus form a convex cone but not a vector space. Furthermore, the total variation defines a norm in respect to which the space of finite signed measures becomes a Banach space. This space has even more structure, in that it can be shown to be a Dedekind complete Banach lattice and in so doing the Radon–Nikodym theorem can be shown to be a special case of the Freudenthal spectral theorem.
If X is a compact separable space, then the space of finite signed Baire measures is the dual of the real Banach space of all continuous real-valued functions on X, by the Riesz representation theorem.
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