Hahn decomposition theorem

The Hahn decomposition theorem states that every signed measure admits a partition into a positive set and a negative set. Any two such decompositions agree up to sets that are null for the signed measure.

Signed measure: A countably additive set function that may take positive, negative, or infinite values, but not both infinities. The theorem applies to signed measures and separates their positive and negative behavior.
Measure space: A set equipped with a sigma-algebra and a measure defined on that sigma-algebra. The theorem partitions the underlying measurable space.
Hans Hahn: An Austrian mathematician known for contributions to functional analysis and measure theory. The decomposition theorem bears his name.
Lebesgue–Stieltjes measure: A measure associated with a nondecreasing function, generalizable to signed measures from functions of bounded variation. Differences of monotone functions produce signed measures whose positive and negative parts can be separated.
Lebesgue decomposition theorem: A theorem splitting a measure into parts absolutely continuous and singular with respect to another measure. It separates by relationship to a reference measure, unlike Hahn's sign-based partition.
Positive set: A measurable set on which every measurable subset has nonnegative signed measure. One half of the partition contains no subset of negative measure.
Sigma-algebra: A collection of subsets closed under complements and countable unions, serving as the domain of a measure. Both pieces of the decomposition must be measurable.
Frigyes Riesz: A Hungarian mathematician whose work shaped functional analysis and modern measure theory. His representation theorems developed alongside the theory of signed measures.
Functions of bounded variation: Functions whose total variation over their domain is finite. Their increments define signed measures, and decomposition isolates increasing contributions.
Jordan decomposition theorem: A theorem expressing a signed measure as the difference of mutually singular positive measures. It decomposes the measure itself into positive measures rather than partitioning its domain.

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