False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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FALSE: convergence in L^1(mu) forces almost-everywhere convergence

Statement refuted

convergence in $L^{1} (μ)$ forces almost-everywhere convergence.

Facts & Assumptions

Given: Lebesgue measure on $[0, 1]$ and the dyadic typewriter sequence $f_{n}$ defined by $f_{0} : = 0$ and $f_{2^{k} + j} : = χ_{I_{k, j}} for k \geq 0, 0 \leq j < 2^{k},$ where $I_{k, j} = [j 2^{- k}, (j + 1) 2^{- k})$ for $j < 2^{k} - 1$ and $I_{k, 2^{k} - 1} = [1 - 2^{- k}, 1]$ .

[L1]

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

[L2]

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

Refutation

technique · direct

1.1givenL1algebra

If $2^{k} \leq n < 2^{k + 1}$ , then $f_{n}$ is the indicator of an interval of length $2^{- k}$ , so $\int_{0}^{1} ∣ f_{n} ∣ d λ = 2^{- k} \to 0.$ Thus $f_{n} \to 0$ in $L^{1} ([0, 1])$ by [L1].

1.2given

Fix $x \in [0, 1]$ . For each $k \geq 1$ there is exactly one $j_{k} \in {0, \dots, 2^{k} - 1}$ with $x \in I_{k, j_{k}}$ , so $f_{2^{k} + j_{k}} (x) = 1$ . Because the same generation contains other dyadic intervals as well, there are also infinitely many indices $n$ with $f_{n} (x) = 0$ . So $(f_{n} (x))$ does not converge for any $x \in [0, 1]$ .

2.1step 1.1step 1.2L2∎

Step 1.1 gives convergence in $L^{1}$ , while step 1.2 shows failure of pointwise convergence at every point and hence failure of [L2]. This refutes the claim.

Depends on

Used by

Dependency tree · two levels

4 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 (iv) (standard reference, not scraped)
Terence Tao, 245A Notes 4: Modes of convergence, Example 7 (standard reference, not scraped)