ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A uniformly integrable family need not admit a single integrable majorant

Example

On $[0, 1]$ with Lebesgue measure, define $a_{k} : = \sum_{j = 1}^{k - 1} \frac{1}{2 j^{2}}, I_{k} : = [a_{k}, a_{k} + \frac{1}{2 k^{2}}), f_{k} : = k χ_{I_{k}} .$

Then the family $(f_{k})_{k \geq 1}$ is uniformly integrable, but no integrable function dominates all of it almost everywhere.

Facts & Assumptions

Given: Lebesgue measure on $[0, 1]$ , the intervals $I_{k}$ , and the functions $f_{k} : = k χ_{I_{k}}$ .

[L1]

Uniform integrability means that for every $ε > 0$ there is $M > 0$ such that $\int_{{∣ f_{k} ∣ > M}} ∣ f_{k} ∣ d μ < ε$ for every $k$ . (A uniformly integrable family)

[L2]

On finite measure spaces, uniform integrability is equivalent to $L^{1}$ -boundedness plus uniform absolute continuity. (On a finite measure space, uniform integrability is equivalent to L^1-boundedness plus uniform absolute continuity)

Verification

technique · direct

1.1givenalgebra

The series $\sum_{k = 1}^{\infty} \frac{1}{2 k^{2}}$ converges to a value below $1$ , so the intervals $I_{k}$ are pairwise disjoint subsets of $[0, 1]$ . Also $\int ∣ f_{k} ∣ d λ = k λ (I_{k}) = \frac{1}{2 k} .$

2.1step 1.1L1algebra

If $M > 0$ and $k \leq M$ , then ${∣ f_{k} ∣ > M} = \emptyset$ ; if $k > M$ , then ${∣ f_{k} ∣ > M} = I_{k}$ and $\int_{{∣ f_{k} ∣ > M}} ∣ f_{k} ∣ d λ = \frac{1}{2 k} \leq \frac{1}{2 ⌊ M ⌋ + 2} .$ Hence the family is uniformly integrable by [L1].

2.2step 1.1algebra

If $g$ were an integrable majorant for all $f_{k}$ , then $∣ g ∣ \geq g \geq k$ almost everywhere on $I_{k}$ . Since the intervals are pairwise disjoint, $\int ∣ g ∣ d λ \geq \sum_{k = 1}^{\infty} \int_{I_{k}} ∣ g ∣ d λ \geq \sum_{k = 1}^{\infty} \frac{1}{2 k} = + \infty,$ contradicting integrability of $g$ .

3.1step 2.1step 2.2L2∎

This family is therefore uniformly integrable but has no single integrable majorant. The example is exactly the failure of the false statement paired with [L2].

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