The Hahn-Jordan decomposition is the canonical decomposition of a signed measure into its positive and negative parts. For a signed measure mu on a measurable space (Omega, B), there exist unique positive measures mu^+ and mu^- such that
mu = mu^+ - mu
and mu^+ and mu^- are mutually singular (supported on disjoint sets). The total variation measure is |mu| = mu^+ + mu^-, and the total variation (or norm) is ||mu|| = |mu|(Omega).
This decomposition plays a central role in the study of measure convergence for signed measures, because the positive and negative parts can behave differently under convergence. Key phenomena:
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Mass cancellation: mu_n can converge vaguely while mu_n^+ and mu_n^- individually lose mass that cancels in the limit. Example (Herdegen-Liang-Shelley 3.5): mu_n = delta_0 - delta_{1/n} converges vaguely (and weakly) to the zero measure, but the distribution functions do not converge because positive and negative mass cancel locally.
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Independent escape: the positive and negative parts can independently escape to infinity. The mass preserving condition hierarchy controls this.
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Convergence of parts: vague convergence of mu_n does NOT imply vague convergence of mu_n^± in general. Additional conditions are needed (Proposition 2.7 in Herdegen-Liang-Shelley): vague convergence + no mass lost on compact sets ⇐> convergence of Hahn-Jordan parts.
Key Details
- Uniqueness: mu^+ and mu^- are uniquely determined and mutually singular
- Total variation: ||mu|| = mu^+(Omega) + mu^-(Omega) = |mu|(Omega)
- Signed Radon measure: a finite signed measure whose total variation is inner regular
- For convergence: vague convergence of mu_n and convergence of mu_n^± are related but not equivalent — mass preservation conditions are needed to ensure one implies the other
Textbook References
Measure Theory - Bogachev (Bogachev, 2007)
- Example 8.4.5 (p. 197): Weak convergence is NOT preserved by the Hahn-Jordan decomposition: mu_n = sin(nx)dx ⇒ 0 but |mu_n| has no weak limit; delta_0 - delta_{1/n} ⇒ 0 but |delta_0 - delta_{1/n}| ⇒ 2*delta_0
- Corollary 8.4.8 (p. 198): H-J parts converge weakly precisely when total variation masses converge: mu_alpha ⇒ mu and |mu_alpha|(X) → |mu|(X) implies mu_alpha^+- ⇒ mu^+-
- Example 8.4.6 (p. 197): Le Cam’s example — on compact metric X, mu_n ⇒ 0 but |mu_n| ⇒ mu which is tau-additive but NOT Radon
Infinite Dimensional Analysis (Aliprantis-Border, 2006)
- Definition 10.2 (p. 374): Signed charge (additive, mu(empty)=0, at most one infinite value); charge (nonneg signed charge); signed measure (sigma-additive); measure (nonneg signed measure)
- Theorem 10.53 (p. 397): ba(A) is an AL-space: lattice operations (mu v nu)(A) = sup{mu(B) + nu(A\B) : B subset A}, total variation norm ||mu|| = V_mu = |mu|(X)
- Corollary 10.54 (p. 398): mu^+(A) = sup{mu(B) : B subset A}, mu^-(A) = -inf{mu(B) : B subset A}, |mu|(A) = sup{sum |mu(A_i)| : partition of A}
- Theorem 10.56 (p. 399): ca(A) is a projection band in ba(A); Yosida-Hewitt decomposition ba(A) = ca(A) + [ca(A)]^d separates countably additive from purely finitely additive parts
- Definition 10.59 (p. 401): nu << mu (absolute continuity) iff for each eps > 0 there exists delta > 0 such that |mu|(A) < delta implies |nu(A)| < eps
- Theorem 10.61 (p. 401): Lebesgue decomposition — every nu in ba(A) uniquely decomposes as nu = nu_1 + nu_2 with nu_1 << mu and nu_2 perp mu