CounterexampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-31

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A nonzero function on a null set has zero $L^{p}$ seminorm

Statement refuted

Every nonzero measurable function has positive $L^{p}$ seminorm.

Facts & Assumptions

Given: The rational indicator on $[0, 1]$ .

[L1]

$Q$ is countable and countable subsets of $R$ are null ( $Q$ is countably infinite, Every at most countable subset of $R$ has measure zero).

[L2]

Null functions are exactly the zero-seminorm class in every range treated on this page (Null functions form a linear subspace and are exactly the zero-seminorm class).

Counterexample

Proof technique: Use the indicator of a countable null subset of $[0, 1]$ . It is not the zero function, but its $p$ -seminorm and essential supremum both vanish.

1.1

Let $f : = χ_{Q \cap [0, 1]}$ . Then $f$ is not the zero function, [given] because $f (q) = 1$ for every rational $q \in [0, 1]$ .

1.2

By [L1], the support of $f$ is null, so $f = 0$ almost everywhere. Therefore [L1, L2] [L2] gives $∥ f ∥_{p} = 0$ in every finite- $p$ range treated on the page, and also $∥ f ∥_{\infty} = 0$ .

2.1

Thus a nonzero measurable function can have zero $L^{p}$ seminorm.

step 1.1step 1.2∎

Depends on

Used by

A Cauchy sequence in calligraphic Lᵖ can converge to two distinct functions Counterexample
FALSE: the p-seminorm on calligraphic Lᵖ is a norm False statement

Dependency tree · two levels

41 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)
Gerald B. Folland, Real Analysis, 2nd ed., Section 2.5 (standard reference, not scraped)

A nonzero function on a null set has zero LpLp seminorm