A signed measure on a measurable space is a set function that is countably additive, vanishes on the empty set, and does not take both and as values (Hunter 2011, pp. 70-71).

A signed measure is finite when all its values are finite. A finite signed measure is a real measure, equivalently a complex measure whose values are all real numbers (Rudin 1987, p. 119).

For a finite signed measure, the Jordan measure decomposition gives a unique representation

where and are finite mutually singular positive measures. Its total variation is , so (Hunter 2011, pp. 73-74). Conversely, a signed measure with finite total variation is finite.

See also

Absolutely Continuous, Bounded Variation, Complex Measure, Jordan Measure Decomposition, Measure, Positive Measure, Real Measure, Total Variation

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References

Hunter, J. K. Measure Theory. Lecture notes, pp. 70-74, 2011. https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes.pdf.Rudin, W. Real and Complex Analysis, 3rd ed. New York: McGraw-Hill, pp. 118-119, 1987.

Cite this as:

Weisstein, Eric W. "Signed Measure." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SignedMeasure.html

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