CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30

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Finite partitions need not attain complex total variation

Statement refuted

For every complex measure, some finite measurable partition attains the total-variation supremum.

Facts & Assumptions

Given: The complex measure $ν (E) = \int_{E} e^{i x} d λ$ on $[0, 2 π]$ .

[L1]

For this measure, $∣ ν ∣ ([0, 2 π]) = 2 π$ . (A complex L^1 density defines a complex measure whose total variation is |h| dmu)

[A1]

If $A \subseteq [0, 2 π]$ has positive Lebesgue measure, then $∣ \int_{A} e^{i x} d λ ∣ < λ (A)$ , because equality in the triangle inequality would force $e^{i x}$ to have constant argument almost everywhere on $A$ .

[A2]

For every $ε > 0$ there is a countable partition of $[0, 2 π]$ into intervals $(I_{n})$ so short that $\sum_{n} ∣ \int_{I_{n}} e^{i x} d λ ∣ > 2 π - ε$ .

Counterexample

technique · direct

1.1

Let $E_{1}, \dots, E_{m}$ be a finite measurable partition of $[0, 2 π]$ . Every [L1, A1] piece of positive measure satisfies the strict inequality from [A1], and the null pieces contribute $0$ . Therefore $\sum_{j = 1}^{m} ∣ ν (E_{j}) ∣ < \sum_{j = 1}^{m} λ (E_{j}) = 2 π .$ So no finite partition attains the total variation value $2 π$ .

L1A1

2.1

By [A2], countable partitions can produce sums arbitrarily close to $2 π$ . [L1, A2, step 1.1] Combining this with step 1.1 and [L1] shows that the total-variation value $2 π$ is not attained by any finite partition. ∎

Depends on

A complex L^1 density defines a complex measure whose total variation is |h| dmu

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sheldon Axler, Measure, Integration & Real Analysis, Chapter 9A (standard reference, not scraped)