Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-27
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FALSE: monotone convergence holds without monotonicity

Statement

If fn:X→[0,+∞] are measurable and converge pointwise to f, then ∫fn dμ→∫f dμ.

Facts & Assumptions

Given: The statement above.

[L1]

Monotone convergence requires a nondecreasing hypothesis (Monotone convergence for the integral).

Refutation

technique · direct
1.1

On (0,1) with Lebesgue measure, let [given, construct] fn:=(n+1)χ(0,1/(n+1)); then fn(x)→0 for every x∈(0,1).

2.1

But ∫fn dμ=1 for every n, whereas ∫0 dμ=0. So the displayed conclusion fails, and [L1] shows that the missing monotonicity hypothesis is exactly what breaks.

step 1.1L1algebra∎

Depends on

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