CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-30

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A finitely additive finite-valued set function can have infinite total variation

Statement refuted

Every finitely additive finite-valued set function has finite total variation.

Facts & Assumptions

Given: The algebra $A$ of finite disjoint unions of half-open intervals $(a, b] \subseteq (0, 1]$ and the function $g (x) = x \sin (1 / x^{2})$ for $x > 0$ , with $g (0) = 0$ .

[A1]

Here "finitely additive" means $ϕ (\emptyset) = 0$ and $ϕ (A \cup B) = ϕ (A) + ϕ (B)$ for disjoint $A, B$ in the domain algebra.

[A2]

Define $ϕ ((a, b]) : = g (b) - g (a)$ and extend by finite additivity to $A$ . Then $ϕ$ is finite-valued on every member of $A$ .

[A3]

For $u_{n} = (2 π n + π / 2)^{- 1 / 2}$ and $v_{n} = (2 π n + 3 π / 2)^{- 1 / 2}$ , one has $g (u_{n}) = u_{n}$ and $g (v_{n}) = - v_{n}$ , so $∣ g (u_{n}) - g (v_{n}) ∣ = u_{n} + v_{n}$ . The series $\sum_{n} (u_{n} + v_{n})$ diverges.

[A4]

For a finitely additive real-valued set function on an algebra, its total variation on $E$ means $∣ ϕ ∣ (E) : = \sup {\sum_{j = 1}^{m} ∣ ϕ (E_{j}) ∣ : E = ⨆_{j = 1}^{m} E_{j}, E_{j} \in A} .$

Counterexample

technique · direct

1.1A1A2

By [A2], the value of $ϕ$ on a finite disjoint union of half-open intervals is the sum of the endpoint increments of $g$ , so [A1] makes $ϕ$ a finitely additive finite-valued set function on $A$ .

2.1A2A3A4∎

For each $N$ , partition the interval $(0, u_{1}]$ by the ordered points $0 < \dots < v_{N} < u_{N} < \dots < v_{1} < u_{1}$ . The resulting finite partition sum for $ϕ$ is at least $\sum_{n = 1}^{N} ∣ g (u_{n}) - g (v_{n}) ∣ = \sum_{n = 1}^{N} (u_{n} + v_{n}) .$ By [A3], these lower bounds diverge with $N$ , so [A4] gives $∣ ϕ ∣ ((0, u_{1}]) = + \infty$ even though every value of $ϕ$ is finite.

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Jordan decomposition and variation of finitely additive charges, standard counterexample family (standard reference, not scraped)