nLab Hahn decomposition theorem


If XX is a set, MM is a σ-algebra on XX, and μ\mu is a signed measure, i.e., a countably additive functional MRM\to\mathbf{R}, then μ\mu is bounded and there is SMS\in M such that

  • μ(m)0\mu(m)\ge0 for every mMm\in M such that mSm\subset S;

  • μ(m)0\mu(m)\le0 for every mMm\in M such that mS=m\cap S=\emptyset.


A standard theorem present in many introductory textbooks. See, for example, Theorem 231E in Fremlin’s Measure Theory.

Created on May 3, 2024 at 03:34:33. See the history of this page for a list of all contributions to it.