TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-08-30

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Integrals against signed or complex measures are bounded by total variation

Statement

Let $ν$ be a signed measure or complex measure on $(X, A)$ and let $f \in L^{1} (ν)$ . Then $∣ \int f d ν ∣ \leq \int ∣ f ∣ d ∣ ν ∣ .$ More generally, for every measurable $E$ , $∣ \int_{E} f d ν ∣ \leq \int_{E} ∣ f ∣ d ∣ ν ∣ .$

Facts & Assumptions

Given: A signed measure or complex measure $ν$ , a function $f \in L^{1} (ν)$ , and a measurable set $E$ .

[L1]

Integration against $ν$ is defined as the limit of simple integrals along an $L^{1} (∣ ν ∣)$ -approximating sequence. (Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|))

[L2]

Simple integrals satisfy $∣ \int_{E} s d ν ∣ \leq \int_{E} ∣ s ∣ d ∣ ν ∣$ . (Simple integrals are bounded by total variation)

Proof

technique · direct

1.1

By [L1], choose complex simple functions $s_{n}$ with [L1, L2] $\int ∣ f - s_{n} ∣ d ∣ ν ∣ \to 0$ and $\int f d ν = \lim_{n} \int s_{n} d ν .$ Applying [L2] to $s_{n} - s_{m}$ shows that $(\int s_{n} d ν)$ is Cauchy.

2.1

By [L2], [L1, L2, step 1.1] $∣ \int s_{n} d ν ∣ \leq \int ∣ s_{n} ∣ d ∣ ν ∣ \leq \int ∣ f ∣ d ∣ ν ∣ + \int ∣ f - s_{n} ∣ d ∣ ν ∣ .$ Letting $n \to \infty$ in step 1.1 yields $∣ \int f d ν ∣ \leq \int ∣ f ∣ d ∣ ν ∣ .$ Applying the same argument to $f 1_{E}$ gives the measurable-subset version.

3.1

Step 2.1 proves both inequalities.

step 2.1∎

Depends on

Used by

A complex L¹ density defines a complex measure whose total variation is |h| dmu Theorem

Dependency tree · two levels

8 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 F. Bass, Real Analysis for Graduate Students, Exercise 12.2 (standard reference, not scraped)
Sheldon Axler, Measure, Integration & Real Analysis, Chapter 9A (standard reference, not scraped)