RemarkRemark: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-31

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$ℓ^{p}$ is the $L^{p}$ space of counting measure

Remark

On $(N, P (N), #)$ with counting measure, every function $f : N \to R$ is measurable. Writing $a_{k} : = f (k)$ , one has

$\int ∣ f ∣^{p} d # = \sum_{k = 0}^{\infty} ∣ a_{k} ∣^{p} (0 < p < \infty),$

by the counting-measure integral dictionary, so $L^{p} (#)$ is exactly the usual sequence class $ℓ^{p}$ . Also

$∥ f ∥_{\infty} = \sup_{k \in N} ∣ a_{k} ∣,$

because a subset of $N$ has counting measure zero only when it is empty. Hence the quotient by almost-everywhere equality does nothing: for counting measure on $N$ , equality almost everywhere means equality everywhere.

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Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Chapter 8 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Example 2.55 and Chapter 7 (standard reference, not scraped)

ℓpℓp is the LpLp space of counting measure