Questions: Signed Measures and Hahn-Jordan Decomposition

Question 1

A signed measure ν on X can be written as ν = μ₁ − μ₂ for many pairs of positive measures. Under what additional condition on μ₁ and μ₂ is this decomposition guaranteed to be unique?

• A. Both μ₁ and μ₂ must be σ-finite
• B. μ₁ and μ₂ must be mutually singular — concentrated on disjoint measurable sets
• C. The total variation μ₁ + μ₂ must be a finite measure
• D. μ₁(X) must equal μ₂(X)

Answer: (B)

Mutual singularity is the key uniqueness condition: ν⁺ concentrates on the Hahn positive set P and ν⁻ concentrates on N = Pᶜ, so they assign zero mass to each other's supporting set. Without this constraint, you could add any positive measure λ to both μ₁ and μ₂ without changing the difference, giving infinitely many decompositions. σ-finiteness and finite total variation are useful regularity conditions but do not enforce uniqueness on their own.

Question 2

The Hahn decomposition partitions X into a positive set P and a negative set N. For every measurable set E ⊆ N, which statement is correct about ν(E)?

• A. ν(E) can be positive or negative depending on E's internal structure
• B. ν(E) ≤ 0 — every measurable subset of N receives nonpositive signed measure
• C. ν(E) = 0 because N contributes no mass to ν
• D. ν(E) is undefined until the Jordan decomposition is computed

Answer: (B)

By definition of the Hahn decomposition, N is a negative set: every measurable subset of N has ν-measure ≤ 0. This is what makes the Jordan decomposition work — ν⁻(E) = −ν(E ∩ N) ≥ 0 for all E. The common misconception is treating N like a null set; N carries negative mass, not zero mass.

Question 3

The total variation measure |ν| = ν⁺ + ν⁻ assigns nonnegative values to all measurable sets, even though ν itself may be negative on some sets.

• T. True
• F. False

Answer: True

True. Since ν⁺ and ν⁻ are both positive measures, their sum |ν| is also a positive measure — it measures the total 'magnitude' of signed mass, analogous to the absolute value of a real number. |ν|(E) = ν⁺(E) + ν⁻(E) ≥ 0 always. This is why |ν| is called the total variation: it captures total mass regardless of sign.

Question 4

The Hahn decomposition of a measurable space into a positive set P and a negative set N is largely unique — there is exactly one such partition with no ambiguity.

• T. True
• F. False

Answer: False

False. The Hahn decomposition is unique only up to ν-null sets. If Z is a set with ν(E) = 0 for all measurable E ⊆ Z (a ν-null set), you can move Z from P to N or vice versa without violating the conditions, producing a different partition that still satisfies all requirements. The decomposition is essentially unique — any two Hahn decompositions differ only on null sets — but not absolutely unique.

Question 5

Explain why the mutual singularity of ν⁺ and ν⁻ in the Jordan decomposition is both necessary for uniqueness and structurally natural given the Hahn decomposition.

Model answer: Mutual singularity means ν⁺ is concentrated on P (the Hahn positive set) and ν⁻ is concentrated on N = Pᶜ. These are disjoint sets, so the two measures have no overlap. This is structurally natural because the positive part of ν 'lives' on the region of X where ν is positive, and the negative part lives where ν is negative. Uniqueness follows: any other decomposition ν = μ₁ − μ₂ into mutually singular positive measures would require μ₁ to concentrate on some set and μ₂ on its complement, which must coincide with P and N (up to null sets). Without mutual singularity, you could shift mass between the two parts, breaking uniqueness.

The Jordan decomposition is canonical precisely because it extracts the intrinsic positive and negative regions of the measure — the Hahn decomposition reveals them. Mutual singularity is the minimal condition that prevents the artificial inflation of both parts by a common positive measure, and it follows directly from the geometry of the Hahn partition.