CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A Cauchy sequence in calligraphic $L^{p}$ can converge to two distinct functions

Statement refuted

In the representative space $L^{p} (μ)$ , the $p$ -distance always gives unique limits.

Facts & Assumptions

Given: A nonzero null-supported function from A nonzero function on a null set has zero $L^{p}$ seminorm.

[L1]

The previous counterexample supplies a measurable function $h \equiv̸ 0$ with $∥ h ∥_{p} = 0$ (A nonzero function on a null set has zero $L^{p}$ seminorm).

[L2]

$L^{p} (μ)$ is the representative function space before quotienting (The function space $L^{p} (μ)$ for $0 < p < \infty$ ).

Counterexample

Proof technique: Take the constant sequence equal to a nonzero function supported on a null set. Its distance to the zero function is $0$ , so it converges to both itself and $0$ in the pseudometric on calligraphic $L^{p}$ .

1.1

Let $f_{n} : = h$ for every $n$ , where $h$ is the function from [L1]. Then [L1, L2] $(f_{n})$ is constant, hence Cauchy in the representative $p$ -distance.

2.1

Also [L1, step 1.1] $∥ f_{n} - h ∥_{p} = 0, ∥ f_{n} - 0 ∥_{p} = ∥ h ∥_{p} = 0$ for every $n$ . So the same sequence converges both to $h$ and to $0$ , even though those two functions are distinct pointwise.

3.1

Therefore the representative-space distance does not have unique limits [step 2.1] before passing to almost-everywhere classes. ∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Section 7B (standard reference, not scraped)

A Cauchy sequence in calligraphic LpLp can converge to two distinct functions