TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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The indefinite integral of a nonnegative measurable function is a measure

Statement

Let $f : X \to [0, + \infty]$ be measurable and define $ν_{f} (A) : = \int_{A} f d μ (A \in A) .$ Then $ν_{f}$ is a measure on $(X, A)$ .

Facts & Assumptions

Given: A nonnegative measurable function $f$ .

[L1]

The set function $A \mapsto \int_{A} f d μ$ is defined by $A \mapsto \int f χ_{A} d μ$ (Integral over a measurable subset).

[L2]

Monotone convergence holds for the nonnegative integral (Monotone convergence for the integral).

[L3]

The nonnegative integral is additive on measurable sets (Additivity of the nonnegative Lebesgue integral).

[L4]

A measure must vanish at the empty set and be countably additive on disjoint measurable sequences (Measures on sigma-algebras).

Proof

technique · direct

1.1L1L2given

One has $ν_{f} (\emptyset) = 0$ . If $(A_{n})$ is a pairwise disjoint sequence, put $B_{n} : = ⋃_{k < n} A_{k}$ . Then $χ_{B_{n}} ↑ χ_{⋃_{k} A_{k}}$ , so $f χ_{B_{n}} ↑ f χ_{⋃_{k} A_{k}}$ , including where $f = + \infty$ under the convention $0 \cdot \infty = 0$ . By [L2], $ν_{f} (⋃_{k} A_{k}) = \lim_{n} ν_{f} (B_{n})$ .

2.1step 1.1L3algebra

Because the sets $A_{k}$ are disjoint, $f χ_{B_{n}} = \sum_{k < n} f χ_{A_{k}}$ . Repeated use of [L3] gives $ν_{f} (B_{n}) = \sum_{k < n} ν_{f} (A_{k})$ . Taking the increasing limit in step 1.1 proves countable additivity.

3.1step 1.1step 2.1L4∎

Steps 1.1 and 2.1 verify the two conditions in [L4], so $ν_{f}$ is a measure.

Depends on

Used by

A C¹ diffeomorphism satisfies the change-of-variables formula for L¹ functions Corollary
Density inversion from an integrable characteristic function Corollary
Distribution of a one-sided Brownian hitting time Corollary
Iid strong law fails at infinite absolute mean Counterexample
Infinite variance can defeat square-root-n CLT scaling Counterexample
Pointwise limit discontinuous at zero signals mass escape Counterexample
Regular conditional laws are not unique on null conditioning values Counterexample
The density ratio is undefined on zero marginal fibres Counterexample
The fundamental Hessian is not absolutely locally integrable Counterexample
Standard normal and normal laws Definition
The measure with density f relative to μ Definition
Weights, their associated measures, and the spaces Lᵖ(w) Definition
Arcsine equilibrium measure and capacity of a segment Example
Bayes formula for a finite mixture with continuous observation Example
Brownian finite-dimensional density Example
Cauchy law and its characteristic function Example
Characteristic function of the uniform law Example
CLT for sums of uniform random variables Example
Conditional density of a bivariate normal law Example
Density inversion for a triangular characteristic function Example
Regular conditional law of one coordinate given another Example
Strong law estimator of an integrable mean Example
The Gauss map preserves Gauss measure Example
The upper half-plane Poisson boundary density Example
Ball and cube maximal functions are pointwise comparable Lemma
Borel change of variables from the compact-support formula and Radon uniqueness Lemma
Maximal dyadic subcubes of a cube at a height Lemma
The Brownian kernels form a semigroup Lemma
The weighted discrete-series space is a Hilbert space with K-type basis Lemma
The indefinite integral of an integrable function is countably additive on measurable sets Proposition
A complex L¹ density defines a complex measure whose total variation is |h| dmu Theorem
A real L¹ density defines a finite signed measure with its canonical Hahn and Jordan data Theorem
Bayes formula for dominated kernels Theorem
Conditional density formula Theorem
Conditional expectation exists by radon nikodym Theorem
Conformal invariance, monotonicity, and the series and parallel laws for extremal length Theorem
Integrating against a density agrees with integrating the product Theorem
Marcinkiewicz interpolation from weak (1,1) and strong (∞,∞) Theorem
Spectral multiplicity model for separably acting abelian von Neumann algebras Theorem
The glued set function is a Borel measure Theorem

Dependency tree · two levels

16 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, ch. 7 (standard reference, not scraped)
Gerald B. Folland, Real Analysis, 2nd ed., §2.2 (standard reference, not scraped)