DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.

The function space $L^{p} (μ)$ for $0 < p < \infty$

Definition

Let $(X, A, μ)$ be a measure space and let $p$ be a real number with $0 < p < \infty$ . Write $M (X, μ)$ for the measurable functions $f : X \to R$ . For $f \in M (X, μ)$ define the extended-valued $p$ -functional

$∥ f ∥_{p} : = {\begin{cases} {(\int ∣ f ∣^{p} d μ)}^{1 / p}, & \int ∣ f ∣^{p} d μ < \infty, \\ + \infty, & \int ∣ f ∣^{p} d μ = + \infty . \end{cases}$

The finite power uses Real powers for positive bases, with the zero-base positive-exponent convention, while the second clause avoids applying real powers to the extended value $+ \infty$ . The integral is the nonnegative Lebesgue integral of The nonnegative Lebesgue integral.

The class

$L^{p} (μ) : = {f \in M (X, μ) : ∥ f ∥_{p} < \infty}$

is the measurable-function space used on this page before passing to almost-everywhere equivalence classes.

For $1 \leq p < \infty$ , the pointwise operations make $L^{p} (μ)$ a real vector space by $L^{p}$ and $L^{\infty}$ are vector spaces for $p \geq 1$ .

The function space $L^{p} (μ)$ for $0 < p < \infty$

Definition

Depends on

Dependency tree · 2 levels

Sources

The function space Lp(μ)Lp(μ) for 0<p<∞0<p<∞

Definition

Depends on

Dependency tree · 2 levels

Sources

The function space $L^{p} (μ)$ for $0 < p < \infty$