A signed measure is a generalization of the concept of a measure, which is a mathematical tool used to assign a size or volume to subsets of a given space, particularly in the context of measure theory. While a traditional measure assigns a non-negative value to subsets, a signed measure allows for the assignment of both positive and negative values.
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A signed measure is a countably additive set function on a sigma-algebra that permits negative values. In the extended-real convention it takes values in , has value zero on the empty set, and does not take both and , so its additive sums are defined. The important finite case is real-valued on every measurable set.[ref]
For distinct points , the difference of Dirac measures is a finite signed measure. It has , and . Its variation measure is , whose total mass is , rather than the absolute value of its signed total mass. The total variation norm of a measure records both contributions before cancellation.
A Hahn decomposition separates a signed measure into a region on which all measurable subsets have nonnegative mass and a region on which all measurable subsets have nonpositive mass. It is the setwise counterpart of splitting an integrable function into its positive part and negative part.