TheoremStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27

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The modulus of an integral is bounded by the integral of the modulus

Statement

If $f \in L^{1} (μ)$ , then $∣ \int f d μ ∣ \leq \int ∣ f ∣ d μ .$

Facts & Assumptions

Given: An integrable function $f$ .

[L1]

The integral is linear on $L^{1} (μ)$ (The Lebesgue integral is linear on $L^{1} (μ)$ ).

[L2]

Real and imaginary parts, complex conjugation, and modulus are as in Real and imaginary parts, complex conjugation, and modulus.

[L3]

Real and complex integrability are defined in Integrable real and complex functions, and their integrals.

[L4]

The nonnegative integral is monotone (Monotonicity and nonnegative homogeneity of the nonnegative integral).

Proof

technique · direct

1.1

For real-valued $f$ , the functions $∣ f ∣ + f = 2 f^{+}$ and $∣ f ∣ - f = 2 f^{-}$ are nonnegative. Therefore [L1] and [L4] give $0 \leq \int (∣ f ∣ + f) d μ = \int ∣ f ∣ d μ + \int f d μ,$ $0 \leq \int (∣ f ∣ - f) d μ = \int ∣ f ∣ d μ - \int f d μ .$ So $- \int ∣ f ∣ d μ \leq \int f d μ \leq \int ∣ f ∣ d μ,$ and hence $∣ \int f d μ ∣ \leq \int ∣ f ∣ d μ$ .

L1L3L4algebra

2.1

For complex-valued $f$ , let $I : = \int f d μ$ . If $I = 0$ there is nothing to prove. Otherwise set $α : = \overline{I} / ∣ I ∣$ , so $∣ α ∣ = 1$ by [L2]. Then $∣ I ∣ = α I = \int α f d μ,$ and $∣ Re (α f) ∣ \leq ∣ α f ∣ = ∣ f ∣$ , so step 1.1 applies to the integrable real-valued function $Re (α f)$ . Taking real parts gives $∣ I ∣ = \int Re (α f) d μ \leq \int ∣ α f ∣ d μ = \int ∣ f ∣ d μ,$ because $Re z \leq ∣ z ∣$ for every complex $z$ .

L1L2step 1.1L4algebra∎

Depends on

Used by

Dominated convergence Theorem

Dependency tree · two levels

11 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

Gerald B. Folland, Real Analysis, 2nd ed., Proposition 2.22 (standard reference, not scraped)
John K. Hunter, Measure Theory Notes, Proposition 4.9 (standard reference, not scraped)