TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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The $p$ -power triangle inequality for nonnegative functions when $0 < p < 1$

Statement

Let $0 < p < 1$ and let $f, g \in L^{p} (μ)$ be nonnegative. Then

$∥ f + g ∥_{p}^{p} \leq ∥ f ∥_{p}^{p} + ∥ g ∥_{p}^{p} .$

Equivalently,

$\int (f + g)^{p} d μ \leq \int f^{p} d μ + \int g^{p} d μ .$

Facts & Assumptions

Given: An exponent $0 < p < 1$ and nonnegative functions $f, g \in L^{p} (μ)$ .

[L1]

For $0 < p < 1$ and nonnegative reals $a, b$ one has $(a + b)^{p} \leq a^{p} + b^{p}$ (The $L^{p}$ distance for $0 < p < 1$ is a complete translation-invariant metric).

[L2]

The nonnegative integral is monotone and additive (Monotonicity and nonnegative homogeneity of the nonnegative integral, Additivity of the nonnegative Lebesgue integral).

Proof

Proof technique: For nonnegative numbers $a, b$ and $0 < p < 1$ one has $(a + b)^{p} \leq a^{p} + b^{p}$ . Apply this pointwise to $f$ and $g$ and integrate.

1.1

The scalar inequality [L1] applied pointwise gives [L1, given] $(f + g)^{p} \leq f^{p} + g^{p} .$

2.1

Integrating and using monotonicity and additivity from [L2] yields [step 1.1, L2] $\int (f + g)^{p} d μ \leq \int f^{p} d μ + \int g^{p} d μ .$ This is exactly the displayed $∥ \cdot ∥_{p}^{p}$ inequality. ∎

Depends on

Used by

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Dependency tree · two levels

21 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

Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Theorem 8.16 (standard reference, not scraped)
John K. Hunter, Measure Theory, reverse inequality discussion before Definition 7.6 (standard reference, not scraped)

The pp-power triangle inequality for nonnegative functions when 0<p<10<p<1