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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The exponential tail function is integrable by monotone truncation and geometric comparison

Example

The function $x \mapsto e^{- x}$ on $[0, \infty)$ is Lebesgue integrable.

Facts & Assumptions

Given: The function $f (x) = e^{- x}$ on $[0, \infty)$ .

[L1]

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

[L2]

Intervals have the expected Lebesgue measure (A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included).

[L3]

The geometric series with ratio $e^{- 1}$ converges (For $∣ r ∣ < 1$ , $\sum_{k \geq 0} r^{k} = 1 / (1 - r)$ , and for $∣ r ∣ \geq 1$ the series diverges).

[L4]

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

Verification

technique · direct

1.1

Let $f_{n} : = e^{- x} χ_{[0, n]}$ . Then $f_{n} ↑ f$ . For each $k \geq 0$ and [L1, L2, construct] $x \in [k, k + 1]$ , one has $e^{- x} \leq e^{- k}$ , so $\int_{[k, k + 1]} f_{n} d λ \leq e^{- k} λ ([k, k + 1]) = e^{- k}$ by [L2].

2.1

Additivity [L4] therefore gives [step 1.1, L1, L3, L4] ∎ $\int f_{n} d λ \leq \sum_{k < n} e^{- k},$ and the right-hand side is bounded independently of $n$ by [L3]. Passing to the limit with [L1] shows $\int_{0}^{\infty} e^{- x} d λ < + \infty$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

37 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

John K. Hunter, Measure Theory Notes, Chapter 4 (standard reference, not scraped)