CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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The spikes k chi_(0,1/k) converge almost everywhere and in measure to zero but not in L^1

Statement refuted

convergence in measure implies convergence in $L^{1} (μ)$ .

Facts & Assumptions

Given: Lebesgue measure on $[0, 1]$ and the sequence defined by $f_{0} : = 0$ and $f_{n} (x) : = n χ_{(0, 1 / n)} (x)$ for $n \geq 1$ .

[L1]

Almost-everywhere convergence means pointwise convergence off a measurable null set. (Convergence almost everywhere relative to a measure)

[L2]

Convergence in measure means that for every real $ε > 0$ , $μ ({∣ f_{n} - f ∣ > ε}) \to 0$ . (Convergence in measure)

[L3]

Convergence in $L^{1} (μ)$ means $\int ∣ f_{n} - f ∣ d μ \to 0$ . (Convergence in L^1(mu))

Counterexample

technique · direct

1.1givenL1

Fix $x \in [0, 1]$ . If $x > 0$ , then $1 / n < x$ for all large $n$ , so $f_{n} (x) = 0$ eventually; and $f_{n} (0) = 0$ for every $n$ . Thus $f_{n} (x) \to 0$ for every $x \in [0, 1]$ , hence almost everywhere by [L1].

1.2givenL2

Fix $ε > 0$ . For all $n > ε$ one has ${∣ f_{n} - 0 ∣ > ε} = (0, 1 / n)$ , whose measure is $1 / n \to 0$ . So $f_{n} \to 0$ in measure by [L2].

1.3

For every $n \geq 1$ , [given, L3, algebra] $\int_{0}^{1} ∣ f_{n} ∣ d λ = \int_{0}^{1 / n} n d λ = 1.$ So the $L^{1}$ errors from $0$ never tend to $0$ , and [L3] fails.

2.1step 1.1step 1.2step 1.3∎

The sequence converges almost everywhere and in measure to $0$ , but not in $L^{1}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Gerald B. Folland, Real Analysis, 2nd ed., Section 2.4, Example (iii) (standard reference, not scraped)