Measure theory and analysis · existing Mathlib declaration

Monotone Convergence Theorem

If measurable nonnegative functions grow pointwise with the sequence index, then integrating their pointwise supremum gives the same extended value as taking the supremum of their integrals.

monotone convergence
Lebesgue integration
nonnegative extended reals
pointwise suprema
measurable functions
lintegrals

The theorem at a glance

Monotone Convergence Theorem at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

An ivory measure-theory poster states the measurable pointwise-monotone hypotheses and the equality between the lintegral of a pointwise supremum and the supremum of the lintegrals. — For measurable ℝ≥0∞-valued functions increasing in pointwise order, lintegral commutes with the pointwise supremum.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

Original 1536 by 1024 pixels inspected. The Scientific figure is visibly distinct from the concept schematic and poster: one shared coordinate construction moves from four ordered measurable profiles, through pointwise fibers selecting the supremum at each location, to nested nonnegative lintegral bands and a common equality endpoint. Every label is permitted and legible, the lintegral and supremum symbols are retained, the possible infinite value is explicit, and there is no signed cancellation, rate, finite-value requirement, or almost-everywhere exception motif. — A pointwise-monotone sequence builds its supremum while the corresponding nonnegative lintegrals rise to the same extended-real value.

Complementary intuition

A mathematical landmark

The monotone convergence theorem is a foundational limit-interchange principle in measure theory. It allows nonnegative mass to be assembled through an increasing sequence without an integrable dominating function and underlies the construction and structural properties of the Lebesgue integral.

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

theorem MeasureTheory.lintegral_iSup {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ

Selected evidence declaration text

theorem MeasureTheory.lintegral_iSup {α : Type*} {m : MeasurableSpace α} {μ : Measure α} {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) : ∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's monotone convergence theorem for a natural-number-indexed sequence f : ℕ → α → ℝ≥0∞ whose terms are measurable and which is monotone in the pointwise function order. It concludes that the lintegral of the pointwise supremum equals the supremum of the lintegrals. Nonnegativity is encoded by ℝ≥0∞, so no integrability or finiteness hypothesis is required and either side may be ∞. The selected declaration does not state the nearby almost-everywhere-measurable/almost-everywhere-monotone variant, a signed or Bochner-integral theorem, convergence for a nonmonotone sequence, or a quantitative rate.

Proof Atlas did not originate the monotone convergence theorem or Mathlib's declaration.
The selected declaration requires ordinary measurability of every term and pointwise monotonicity in the function order, not merely almost-everywhere versions.
The conclusion concerns ℝ≥0∞-valued lintegrals and permits the common value ∞; it is not a signed or Bochner-integral statement.
The generated explanation and visuals are not proof evidence.

Provenance

Origin and evidence stay separate

Existing declaration: MeasureTheory.lintegral_iSup in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.monotone-convergence-theorem.v001
Source handling: Verified upstream reference; no mirrored source package