ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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Finite counting measure on $n$ points recovers $R^{n}$ $p$ -norms

Example

Let $n \geq 1$ , let $X = {0, \dots, n - 1}$ with counting measure, and let $f : X \to R$ be given by $f (k) = x_{k}$ . Then

$∥ f ∥_{p} = {(\sum_{k = 0}^{n - 1} ∣ x_{k} ∣^{p})}^{1 / p} (0 < p < \infty),$

and

$∥ f ∥_{\infty} = \max_{0 \leq k < n} ∣ x_{k} ∣ .$

For every rational $p \geq 1$ , these are exactly the published $p$ -norms on $R^{n}$ from The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ ; the displayed finite-sum formula itself remains valid for every real $p > 0$ .

Facts & Assumptions

Given: An integer $n \geq 1$ , the finite set $X = {0, \dots, n - 1}$ , and a function $f (k) = x_{k}$ .

[L1]

$ℓ^{p}$ is the counting-measure version of $L^{p}$ ( $ℓ^{p}$ is the $L^{p}$ space of counting measure).

[L2]

For rational $p \geq 1$ , The $p$ -norms $∥ x ∥_{p}$ for rational $p \geq 1$ , and $∥ x ∥_{\infty}$ defines the finite-dimensional $p$ -norms by the same finite-sum formula, and for $n \geq 1$ it defines the same maximum norm.

Verification

Proof technique: Unwind the counting-measure integral on a finite set and compare it term by term with the published $p$ -norms on $R^{n}$ .

1.1

Extending $(x_{0}, \dots, x_{n - 1})$ by zeros outside ${0, \dots, n - 1}$ turns it [L1, given] into a sequence in the setting of [L1]. The resulting $ℓ^{p}$ and $L^{\infty}$ formulas are exactly the two displayed expressions.

2.1

In the ranges stated in [L2], those expressions are exactly the published [step 1.1, L2] norms on $R^{n}$ . For other real $p > 0$ , step 1.1 still gives the displayed $L^{p}$ functional, without claiming that the earlier finite-dimensional page called it a norm. ∎

Depends on

Used by

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Sources

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Chapter 8 (standard reference, not scraped)

Finite counting measure on nn points recovers RnRn pp-norms