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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A decreasing sequence need not satisfy a monotone convergence theorem without an integrable start

Statement refuted

If $f_{n} ↓ f$ pointwise for nonnegative measurable functions, then $\int f_{n} d μ ↓ \int f d μ .$

Facts & Assumptions

Given: The decreasing sequence $f_{n} : = χ_{[n, \infty)}$ on $R$ .

[L1]

The monotone convergence theorem is an increasing theorem, not a decreasing one (Monotone convergence for the integral).

Counterexample

technique · direct

1.1

The functions $f_{n}$ decrease pointwise to $0$ .

given

2.1

But $\int f_{n} d λ = + \infty$ for every $n$ , while $\int 0 d λ = 0$ . [step 1.1, L1, algebra] ∎ So the displayed conclusion fails, confirming the directionality recorded in [L1].

Depends on

Monotone convergence for the integral

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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., §2.2 (standard reference, not scraped)